Termination w.r.t. Q proof of /tmp/tmpnfylnu/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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 17 SCCs with 58 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ 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:

QUOTE1(ok(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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

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

          ↳ QDP

          ↳ QDP

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

QUOTE1(ok(X)) → QUOTE1(X)

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:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ 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:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ 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:

QUOTE(ok(X)) → QUOTE(X)

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

QUOTE(ok(X)) → QUOTE(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

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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(ok(X)) → UNQUOTE1(X)
UNQUOTE1(mark(X)) → UNQUOTE1(X)

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

UNQUOTE1(ok(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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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(ok(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(ok(X)) → UNQUOTE(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

          ↳ QDP

          ↳ QDP

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

S1(ok(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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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(ok(X)) → S1(X)
S1(mark(X)) → S1(X)

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

S1(ok(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

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

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

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

CONS1(ok(X1), ok(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(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

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

FIRST1(ok(X1), ok(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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

FIRST1(ok(X1), ok(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(ok(X1), ok(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

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

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

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

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

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

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

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

FROM(mark(X)) → FROM(X)
FROM(ok(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(ok(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

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

FIRST(ok(X1), ok(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(ok(X1), ok(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

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

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

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

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

          ↳ QDP

          ↳ QDP

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

S(ok(X)) → S(X)
S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

S(ok(X)) → S(X)
S(mark(X)) → S(X)

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

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

          ↳ QDP

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

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

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

PROPER(cons1(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(sel(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(first(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(cons1(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(unquote(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(quote(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(first(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(s(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(sel1(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(fcons(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(quote1(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(sel1(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(sel(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(cons(X1, X2)) → PROPER(X2)
The graph contains the following edges 1 > 1
PROPER(fcons(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(from(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(s1(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(unquote1(X)) → PROPER(X)
The graph contains the following edges 1 > 1
PROPER(first1(X1, X2)) → PROPER(X1)
The graph contains the following edges 1 > 1
PROPER(first1(X1, X2)) → PROPER(X2)
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

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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:

ACTIVE(fcons(X1, X2)) → ACTIVE(X2)
ACTIVE(unquote1(X)) → ACTIVE(X)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(s1(X)) → ACTIVE(X)
ACTIVE(sel1(X1, X2)) → ACTIVE(X1)
ACTIVE(cons1(X1, X2)) → ACTIVE(X1)
ACTIVE(unquote(X)) → ACTIVE(X)
ACTIVE(first1(X1, X2)) → ACTIVE(X2)
ACTIVE(fcons(X1, X2)) → ACTIVE(X1)
ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(sel1(X1, X2)) → ACTIVE(X2)
ACTIVE(cons1(X1, X2)) → ACTIVE(X2)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(first(X1, X2)) → ACTIVE(X1)
ACTIVE(first(X1, X2)) → ACTIVE(X2)
ACTIVE(first1(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)

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

ACTIVE(fcons(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(unquote1(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(sel(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(s1(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(sel1(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(cons1(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(unquote(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(first1(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(fcons(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(s(X)) → ACTIVE(X)
The graph contains the following edges 1 > 1
ACTIVE(sel1(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(cons1(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(first(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(first(X1, X2)) → ACTIVE(X2)
The graph contains the following edges 1 > 1
ACTIVE(first1(X1, X2)) → ACTIVE(X1)
The graph contains the following edges 1 > 1
ACTIVE(from(X)) → ACTIVE(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

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the usable rules with reduction pair processor [15] with a polynomial ordering [25], all dependency pairs and the corresponding usable rules [17] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [25]:

POL(0) = 0
POL(01) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = 2·x₁
POL(cons(x₁, x₂)) = 2·x₁ + x₂
POL(cons1(x₁, x₂)) = x₁ + x₂
POL(fcons(x₁, x₂)) = 2·x₁ + x₂
POL(first(x₁, x₂)) = x₁ + 2·x₂
POL(first1(x₁, x₂)) = 2·x₁ + 2·x₂
POL(from(x₁)) = 2·x₁
POL(mark(x₁)) = x₁
POL(nil) = 0
POL(nil1) = 0
POL(ok(x₁)) = 2·x₁
POL(proper(x₁)) = x₁
POL(quote(x₁)) = x₁
POL(quote1(x₁)) = 2·x₁
POL(s(x₁)) = x₁
POL(s1(x₁)) = 2·x₁
POL(sel(x₁, x₂)) = 2·x₁ + x₂
POL(sel1(x₁, x₂)) = 2·x₁ + 2·x₂
POL(unquote(x₁)) = x₁
POL(unquote1(x₁)) = x₁

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

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

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(mark(X)) → TOP(proper(X)) at position [0] we obtained the following new rules:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

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

TOP(mark(quote1(x0))) → TOP(quote1(proper(x0)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(nil1)) → TOP(ok(nil1))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(mark(0)) → TOP(ok(0))
TOP(mark(01)) → TOP(ok(01))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(nil)) → TOP(ok(nil))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(ok(X)) → TOP(active(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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule TOP(ok(X)) → TOP(active(X)) at position [0] we obtained the following new rules:

TOP(ok(quote(0))) → TOP(mark(01))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(ok(quote(sel(x0, x1)))) → TOP(mark(sel1(x0, x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(unquote(01))) → TOP(mark(0))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(first(0, x0))) → TOP(mark(nil))
TOP(ok(quote1(first(x0, x1)))) → TOP(mark(first1(x0, x1)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(ok(quote1(nil))) → TOP(mark(nil1))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(unquote1(nil1))) → TOP(mark(nil))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(quote1(cons(x0, x1)))) → TOP(mark(cons1(quote(x0), quote1(x1))))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(first1(0, x0))) → TOP(mark(nil1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

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

TOP(ok(quote(0))) → TOP(mark(01))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(first(0, x0))) → TOP(mark(nil))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(mark(01)) → TOP(ok(01))
TOP(mark(0)) → TOP(ok(0))
TOP(ok(unquote1(nil1))) → TOP(mark(nil))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))
TOP(ok(first1(0, x0))) → TOP(mark(nil1))
TOP(mark(quote1(x0))) → TOP(quote1(proper(x0)))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(quote(sel(x0, x1)))) → TOP(mark(sel1(x0, x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(mark(nil)) → TOP(ok(nil))
TOP(ok(unquote(01))) → TOP(mark(0))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(quote1(first(x0, x1)))) → TOP(mark(first1(x0, x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(quote1(nil))) → TOP(mark(nil1))
TOP(mark(nil1)) → TOP(ok(nil1))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(quote1(cons(x0, x1)))) → TOP(mark(cons1(quote(x0), quote1(x1))))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, 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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 10 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(quote1(x0))) → TOP(quote1(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(quote(sel(x0, x1)))) → TOP(mark(sel1(x0, x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(quote1(first(x0, x1)))) → TOP(mark(first1(x0, x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(quote1(cons(x0, x1)))) → TOP(mark(cons1(quote(x0), quote1(x1))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, 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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(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.

TOP(ok(quote1(first(x0, x1)))) → TOP(mark(first1(x0, x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(quote1(x0))) → TOP(quote1(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(quote(sel(x0, x1)))) → TOP(mark(sel1(x0, x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(quote1(cons(x0, x1)))) → TOP(mark(cons1(quote(x0), quote1(x1))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

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

TOP(mark(quote1(x0))) → TOP(quote1(proper(x0)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(quote(sel(x0, x1)))) → TOP(mark(sel1(x0, x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(quote1(cons(x0, x1)))) → TOP(mark(cons1(quote(x0), quote1(x1))))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))

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

TOP(ok(quote1(cons(x0, x1)))) → TOP(mark(cons1(quote(x0), quote1(x1))))

The remaining pairs can at least be oriented weakly.

TOP(mark(quote1(x0))) → TOP(quote1(proper(x0)))
TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(quote(sel(x0, x1)))) → TOP(mark(sel1(x0, x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(mark(quote1(x0))) → TOP(quote1(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(quote(sel(x0, x1)))) → TOP(mark(sel1(x0, x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, 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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(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.

TOP(mark(quote1(x0))) → TOP(quote1(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(quote(sel(x0, x1)))) → TOP(mark(sel1(x0, x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(01) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 1
POL(cons1(x₁, x₂)) = 1
POL(fcons(x₁, x₂)) = 1
POL(first(x₁, x₂)) = 1
POL(first1(x₁, x₂)) = 1
POL(from(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(nil1) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 1
POL(quote(x₁)) = x₁
POL(quote1(x₁)) = 0
POL(s(x₁)) = 1
POL(s1(x₁)) = 1
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:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(quote(sel(x0, x1)))) → TOP(mark(sel1(x0, x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))

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

TOP(ok(quote(sel(x0, x1)))) → TOP(mark(sel1(x0, x1)))
TOP(ok(sel(0, cons(x0, x1)))) → TOP(mark(x0))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, 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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(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.

TOP(ok(unquote(s1(x0)))) → TOP(mark(s(unquote(x0))))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))

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

TOP(ok(first1(s(x0), cons(x1, x2)))) → TOP(mark(cons1(quote(x1), first1(x0, x2))))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, 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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(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.

TOP(ok(first(s(x0), cons(x1, x2)))) → TOP(mark(cons(x1, first(x0, x2))))
TOP(ok(from(x0))) → TOP(mark(cons(x0, from(s(x0)))))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))

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

TOP(ok(unquote1(cons1(x0, x1)))) → TOP(mark(fcons(unquote(x0), unquote1(x1))))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, 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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(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.

TOP(ok(fcons(x0, x1))) → TOP(mark(cons(x0, x1)))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, 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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(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.

TOP(ok(quote(s(x0)))) → TOP(mark(s1(quote(x0))))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, 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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(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.

TOP(ok(sel1(0, cons(x0, x1)))) → TOP(mark(quote(x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))

Used ordering: Polynomial interpretation [25]:

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

The following usable rules [17] were oriented:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(quote(x0))) → TOP(quote(proper(x0)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, 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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(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.

TOP(mark(quote(x0))) → TOP(quote(proper(x0)))

The remaining pairs can at least be oriented weakly.

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, x2)))

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(01) = 0
POL(TOP(x₁)) = x₁
POL(active(x₁)) = 0
POL(cons(x₁, x₂)) = 1
POL(cons1(x₁, x₂)) = 1
POL(fcons(x₁, x₂)) = 1
POL(first(x₁, x₂)) = 1
POL(first1(x₁, x₂)) = 1
POL(from(x₁)) = 1
POL(mark(x₁)) = 1
POL(nil) = 0
POL(nil1) = 0
POL(ok(x₁)) = x₁
POL(proper(x₁)) = 0
POL(quote(x₁)) = 0
POL(quote1(x₁)) = 0
POL(s(x₁)) = 1
POL(s1(x₁)) = 1
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:

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesReductionPairsProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

                        ↳ DependencyGraphProof

                          ↳ QDP

                            ↳ QDPOrderProof

                              ↳ QDP

                                ↳ QDPOrderProof

                                  ↳ QDP

                                    ↳ QDPOrderProof

                                      ↳ QDP

                                        ↳ QDPOrderProof

                                          ↳ QDP

                                            ↳ QDPOrderProof

                                              ↳ QDP

                                                ↳ QDPOrderProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ QDPOrderProof

                                                          ↳ QDP

                                                            ↳ QDPOrderProof

                                                              ↳ QDP

                                                                ↳ QDPOrderProof

                                                                  ↳ QDP

                                                                    ↳ QDPOrderProof

                                                                      ↳ QDP

                                                                        ↳ QDPOrderProof

                                                                          ↳ QDP

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

TOP(mark(from(x0))) → TOP(from(proper(x0)))
TOP(ok(first1(x0, x1))) → TOP(first1(x0, active(x1)))
TOP(mark(first(x0, x1))) → TOP(first(proper(x0), proper(x1)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(x0, active(x1)))
TOP(mark(unquote1(x0))) → TOP(unquote1(proper(x0)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(active(x0), x1))
TOP(ok(first1(x0, x1))) → TOP(first1(active(x0), x1))
TOP(ok(unquote(x0))) → TOP(unquote(active(x0)))
TOP(ok(unquote1(x0))) → TOP(unquote1(active(x0)))
TOP(mark(first1(x0, x1))) → TOP(first1(proper(x0), proper(x1)))
TOP(mark(fcons(x0, x1))) → TOP(fcons(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(x0, active(x1)))
TOP(ok(cons(x0, x1))) → TOP(cons(active(x0), x1))
TOP(ok(sel(s(x0), cons(x1, x2)))) → TOP(mark(sel(x0, x2)))
TOP(mark(unquote(x0))) → TOP(unquote(proper(x0)))
TOP(ok(first(x0, x1))) → TOP(first(x0, active(x1)))
TOP(ok(from(x0))) → TOP(from(active(x0)))
TOP(ok(sel(x0, x1))) → TOP(sel(x0, active(x1)))
TOP(ok(s1(x0))) → TOP(s1(active(x0)))
TOP(mark(s(x0))) → TOP(s(proper(x0)))
TOP(ok(sel1(x0, x1))) → TOP(sel1(active(x0), x1))
TOP(ok(sel(x0, x1))) → TOP(sel(active(x0), x1))
TOP(ok(s(x0))) → TOP(s(active(x0)))
TOP(mark(sel(x0, x1))) → TOP(sel(proper(x0), proper(x1)))
TOP(mark(cons1(x0, x1))) → TOP(cons1(proper(x0), proper(x1)))
TOP(ok(cons1(x0, x1))) → TOP(cons1(x0, active(x1)))
TOP(ok(first(x0, x1))) → TOP(first(active(x0), x1))
TOP(mark(cons(x0, x1))) → TOP(cons(proper(x0), proper(x1)))
TOP(mark(s1(x0))) → TOP(s1(proper(x0)))
TOP(mark(sel1(x0, x1))) → TOP(sel1(proper(x0), proper(x1)))
TOP(ok(fcons(x0, x1))) → TOP(fcons(active(x0), x1))
TOP(ok(sel1(s(x0), cons(x1, x2)))) → TOP(mark(sel1(x0, 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))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
unquote1(mark(X)) → mark(unquote1(X))
unquote1(ok(X)) → ok(unquote1(X))
unquote(mark(X)) → mark(unquote(X))
unquote(ok(X)) → ok(unquote(X))
s1(mark(X)) → mark(s1(X))
s1(ok(X)) → ok(s1(X))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
from(mark(X)) → mark(from(X))
from(ok(X)) → ok(from(X))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
cons(mark(X1), X2) → mark(cons(X1, X2))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
s(mark(X)) → mark(s(X))
s(ok(X)) → ok(s(X))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
quote(ok(X)) → ok(quote(X))
quote1(ok(X)) → ok(quote1(X))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))

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