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

(0) Obligation:

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

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

(1) DependencyPairsProof (EQUIVALENT transformation)

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

(2) Obligation:

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

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

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

(3) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 17 SCCs with 58 less nodes.

(4) Complex Obligation (AND)

(5) Obligation:

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.

(6) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QUOTE1(ok(X)) → QUOTE1(X)

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

ok₁ > QUOTE1₁

Status:

ok₁: multiset
QUOTE1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(7) Obligation:

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

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

(8) PisEmptyProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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))

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

(11) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

QUOTE(ok(X)) → QUOTE(X)

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

ok₁ > QUOTE₁

Status:

ok₁: multiset
QUOTE₁: multiset

The following usable rules [FROCOS05] were oriented: none

(12) Obligation:

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

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

(13) PisEmptyProof (EQUIVALENT transformation)

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

(14) TRUE

(15) Obligation:

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

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

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

(16) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

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

trivial

Status:

mark₁: multiset
FCONS₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(17) Obligation:

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

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))

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

(18) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FCONS(ok(X1), ok(X2)) → FCONS(X1, X2)

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

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

[FCONS₁, ok₁]

Status:

ok₁: [1]
FCONS₁: multiset

The following usable rules [FROCOS05] were oriented: none

(19) Obligation:

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

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

(20) PisEmptyProof (EQUIVALENT transformation)

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

(21) TRUE

(22) Obligation:

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))

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

(23) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE1(ok(X)) → UNQUOTE1(X)

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

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

[UNQUOTE1₁, ok₁]

Status:

ok₁: multiset
UNQUOTE1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(24) Obligation:

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

UNQUOTE1(mark(X)) → UNQUOTE1(X)

The TRS R consists of the following rules:

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

(25) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE1(mark(X)) → UNQUOTE1(X)

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

mark₁ > UNQUOTE1₁

Status:

mark₁: multiset
UNQUOTE1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(26) Obligation:

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

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

(27) PisEmptyProof (EQUIVALENT transformation)

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

(28) TRUE

(29) Obligation:

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

UNQUOTE(ok(X)) → UNQUOTE(X)
UNQUOTE(mark(X)) → UNQUOTE(X)

The TRS R consists of the following rules:

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

(30) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE(ok(X)) → UNQUOTE(X)

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

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

[UNQUOTE₁, ok₁]

Status:

ok₁: multiset
UNQUOTE₁: multiset

The following usable rules [FROCOS05] were oriented: none

(31) Obligation:

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

UNQUOTE(mark(X)) → UNQUOTE(X)

The TRS R consists of the following rules:

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

(32) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

UNQUOTE(mark(X)) → UNQUOTE(X)

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

mark₁ > UNQUOTE₁

Status:

mark₁: multiset
UNQUOTE₁: multiset

The following usable rules [FROCOS05] were oriented: none

(33) Obligation:

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

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

(34) PisEmptyProof (EQUIVALENT transformation)

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

(35) TRUE

(36) Obligation:

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))

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

(37) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S1(ok(X)) → S1(X)

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

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

[S1₁, ok₁]

Status:

S1₁: multiset
ok₁: multiset

The following usable rules [FROCOS05] were oriented: none

(38) Obligation:

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

S1(mark(X)) → S1(X)

The TRS R consists of the following rules:

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

(39) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S1(mark(X)) → S1(X)

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

mark₁ > S1₁

Status:

S1₁: multiset
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(40) Obligation:

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

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

(41) PisEmptyProof (EQUIVALENT transformation)

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

(42) TRUE

(43) Obligation:

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

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

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

(44) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

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

trivial

Status:

CONS1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(45) Obligation:

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

CONS1(ok(X1), ok(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))

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

(46) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS1(ok(X1), ok(X2)) → CONS1(X1, X2)

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

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

[CONS1₁, ok₁]

Status:

CONS1₁: multiset
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(47) Obligation:

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

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

(48) PisEmptyProof (EQUIVALENT transformation)

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

(49) TRUE

(50) Obligation:

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

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

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

(51) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

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

trivial

Status:

FIRST1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(52) Obligation:

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

FIRST1(ok(X1), ok(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))

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

(53) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST1(ok(X1), ok(X2)) → FIRST1(X1, X2)

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

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

[FIRST1₁, ok₁]

Status:

FIRST1₁: multiset
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(54) Obligation:

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

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

(55) PisEmptyProof (EQUIVALENT transformation)

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

(56) TRUE

(57) Obligation:

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

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

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

(58) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

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

trivial

Status:

SEL1₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(59) Obligation:

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

SEL1(ok(X1), ok(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))

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

(60) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL1(ok(X1), ok(X2)) → SEL1(X1, X2)

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

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

[SEL1₁, ok₁]

Status:

SEL1₁: multiset
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(61) Obligation:

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

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

(62) PisEmptyProof (EQUIVALENT transformation)

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

(63) TRUE

(64) Obligation:

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

FROM(ok(X)) → FROM(X)
FROM(mark(X)) → FROM(X)

The TRS R consists of the following rules:

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

(65) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROM(ok(X)) → FROM(X)

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

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

[FROM₁, ok₁]

Status:

ok₁: multiset
FROM₁: multiset

The following usable rules [FROCOS05] were oriented: none

(66) Obligation:

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

FROM(mark(X)) → FROM(X)

The TRS R consists of the following rules:

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

(67) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FROM(mark(X)) → FROM(X)

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

mark₁ > FROM₁

Status:

mark₁: multiset
FROM₁: multiset

The following usable rules [FROCOS05] were oriented: none

(68) Obligation:

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

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

(69) PisEmptyProof (EQUIVALENT transformation)

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

(70) TRUE

(71) Obligation:

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

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

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

(72) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

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

trivial

Status:

FIRST₂: [2,1]
mark₁: multiset

The following usable rules [FROCOS05] were oriented: none

(73) Obligation:

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

FIRST(ok(X1), ok(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))

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

(74) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FIRST(ok(X1), ok(X2)) → FIRST(X1, X2)

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

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

[FIRST₁, ok₁]

Status:

FIRST₁: multiset
ok₁: [1]

The following usable rules [FROCOS05] were oriented: none

(75) Obligation:

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

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

(76) PisEmptyProof (EQUIVALENT transformation)

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

(77) TRUE

(78) Obligation:

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

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

The TRS R consists of the following rules:

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

(79) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(ok(X1), ok(X2)) → CONS(X1, X2)

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

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

ok₁ > CONS₁
mark > CONS₁

Status:

CONS₁: multiset
mark: multiset
ok₁: multiset

The following usable rules [FROCOS05] were oriented: none

(80) Obligation:

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

CONS(mark(X1), X2) → CONS(X1, X2)

The TRS R consists of the following rules:

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

(81) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

CONS(mark(X1), X2) → CONS(X1, X2)

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

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

[CONS₁, mark₁]

Status:

CONS₁: multiset
mark₁: [1]

The following usable rules [FROCOS05] were oriented: none

(82) Obligation:

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

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

(83) PisEmptyProof (EQUIVALENT transformation)

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

(84) TRUE

(85) Obligation:

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))

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

(86) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(ok(X)) → S(X)

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

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

[S₁, ok₁]

Status:

ok₁: multiset
S₁: multiset

The following usable rules [FROCOS05] were oriented: none

(87) Obligation:

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

S(mark(X)) → S(X)

The TRS R consists of the following rules:

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

(88) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

S(mark(X)) → S(X)

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

mark₁ > S₁

Status:

mark₁: multiset
S₁: multiset

The following usable rules [FROCOS05] were oriented: none

(89) Obligation:

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

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

(90) PisEmptyProof (EQUIVALENT transformation)

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

(91) TRUE

(92) Obligation:

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

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

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

(93) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

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

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

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

trivial

Status:

mark₁: multiset
SEL₂: [2,1]

The following usable rules [FROCOS05] were oriented: none

(94) Obligation:

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

SEL(ok(X1), ok(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))

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

(95) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

SEL(ok(X1), ok(X2)) → SEL(X1, X2)

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

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

[SEL₁, ok₁]

Status:

ok₁: [1]
SEL₁: multiset

The following usable rules [FROCOS05] were oriented: none

(96) Obligation:

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

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

(97) PisEmptyProof (EQUIVALENT transformation)

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

(98) TRUE

(99) Obligation:

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

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

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

(100) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(sel(X1, X2)) → PROPER(X2)
PROPER(sel(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X1)
PROPER(cons(X1, X2)) → PROPER(X2)
PROPER(first(X1, X2)) → PROPER(X1)
PROPER(first(X1, X2)) → PROPER(X2)
PROPER(sel1(X1, X2)) → PROPER(X1)
PROPER(sel1(X1, X2)) → PROPER(X2)
PROPER(first1(X1, X2)) → PROPER(X1)
PROPER(first1(X1, X2)) → PROPER(X2)
PROPER(cons1(X1, X2)) → PROPER(X1)
PROPER(cons1(X1, X2)) → PROPER(X2)
PROPER(fcons(X1, X2)) → PROPER(X1)
PROPER(fcons(X1, X2)) → PROPER(X2)

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

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

[PROPER₁, first₂, cons1₂, fcons₂]

Status:

sel1₂: multiset
sel₂: multiset
cons₂: multiset
PROPER₁: [1]
cons1₂: multiset
fcons₂: multiset
first1₂: multiset
first₂: multiset

The following usable rules [FROCOS05] were oriented: none

(101) Obligation:

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

PROPER(s(X)) → PROPER(X)
PROPER(from(X)) → PROPER(X)
PROPER(quote(X)) → PROPER(X)
PROPER(quote1(X)) → PROPER(X)
PROPER(s1(X)) → PROPER(X)
PROPER(unquote(X)) → PROPER(X)
PROPER(unquote1(X)) → PROPER(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.

(102) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(s(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
s(x1)  =  s(x1)
from(x1)  =  x1
quote(x1)  =  x1
quote1(x1)  =  x1
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1

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

[PROPER₁, s₁]

Status:

PROPER₁: multiset
s₁: [1]

The following usable rules [FROCOS05] were oriented: none

(103) Obligation:

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

PROPER(from(X)) → PROPER(X)
PROPER(quote(X)) → PROPER(X)
PROPER(quote1(X)) → PROPER(X)
PROPER(s1(X)) → PROPER(X)
PROPER(unquote(X)) → PROPER(X)
PROPER(unquote1(X)) → PROPER(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.

(104) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(unquote1(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
from(x1)  =  x1
quote(x1)  =  x1
quote1(x1)  =  x1
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  unquote1(x1)

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

trivial

Status:

unquote1₁: multiset
PROPER₁: multiset

The following usable rules [FROCOS05] were oriented: none

(105) Obligation:

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

PROPER(from(X)) → PROPER(X)
PROPER(quote(X)) → PROPER(X)
PROPER(quote1(X)) → PROPER(X)
PROPER(s1(X)) → PROPER(X)
PROPER(unquote(X)) → PROPER(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.

(106) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(from(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
from(x1)  =  from(x1)
quote(x1)  =  x1
quote1(x1)  =  x1
s1(x1)  =  x1
unquote(x1)  =  x1

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

[PROPER₁, from₁]

Status:

from₁: multiset
PROPER₁: multiset

The following usable rules [FROCOS05] were oriented: none

(107) Obligation:

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

PROPER(quote(X)) → PROPER(X)
PROPER(quote1(X)) → PROPER(X)
PROPER(s1(X)) → PROPER(X)
PROPER(unquote(X)) → PROPER(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.

(108) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(s1(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
quote(x1)  =  x1
quote1(x1)  =  x1
s1(x1)  =  s1(x1)
unquote(x1)  =  x1

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

[PROPER₁, s1₁]

Status:

s1₁: multiset
PROPER₁: multiset

The following usable rules [FROCOS05] were oriented: none

(109) Obligation:

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

PROPER(quote(X)) → PROPER(X)
PROPER(quote1(X)) → PROPER(X)
PROPER(unquote(X)) → PROPER(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.

(110) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(quote(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
quote(x1)  =  quote(x1)
quote1(x1)  =  x1
unquote(x1)  =  x1

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

[PROPER₁, quote₁]

Status:

PROPER₁: multiset
quote₁: multiset

The following usable rules [FROCOS05] were oriented: none

(111) Obligation:

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

PROPER(quote1(X)) → PROPER(X)
PROPER(unquote(X)) → PROPER(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.

(112) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(quote1(X)) → PROPER(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
PROPER(x1)  =  PROPER(x1)
quote1(x1)  =  quote1(x1)
unquote(x1)  =  x1

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

[PROPER₁, quote1₁]

Status:

PROPER₁: multiset
quote1₁: multiset

The following usable rules [FROCOS05] were oriented: none

(113) Obligation:

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

PROPER(unquote(X)) → PROPER(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.

(114) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

PROPER(unquote(X)) → PROPER(X)

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

unquote₁ > PROPER₁

Status:

PROPER₁: multiset
unquote₁: multiset

The following usable rules [FROCOS05] were oriented: none

(115) Obligation:

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

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

(116) PisEmptyProof (EQUIVALENT transformation)

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

(117) TRUE

(118) Obligation:

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

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

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

(119) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(sel(X1, X2)) → ACTIVE(X2)
ACTIVE(sel(X1, X2)) → ACTIVE(X1)
ACTIVE(first(X1, X2)) → ACTIVE(X1)
ACTIVE(first(X1, X2)) → ACTIVE(X2)
ACTIVE(sel1(X1, X2)) → ACTIVE(X1)
ACTIVE(sel1(X1, X2)) → ACTIVE(X2)
ACTIVE(first1(X1, X2)) → ACTIVE(X1)
ACTIVE(first1(X1, X2)) → ACTIVE(X2)
ACTIVE(cons1(X1, X2)) → ACTIVE(X1)
ACTIVE(cons1(X1, X2)) → ACTIVE(X2)
ACTIVE(fcons(X1, X2)) → ACTIVE(X1)
ACTIVE(fcons(X1, X2)) → ACTIVE(X2)

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

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

sel₂ > [ACTIVE₁, fcons₂]
first₂ > [ACTIVE₁, fcons₂]
sel1₂ > [ACTIVE₁, fcons₂]
first1₂ > [ACTIVE₁, fcons₂]
cons1₂ > [ACTIVE₁, fcons₂]

Status:

sel1₂: multiset
sel₂: multiset
fcons₂: multiset
cons1₂: multiset
first1₂: multiset
first₂: [1,2]
ACTIVE₁: multiset

The following usable rules [FROCOS05] were oriented: none

(120) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(s1(X)) → ACTIVE(X)
ACTIVE(unquote(X)) → ACTIVE(X)
ACTIVE(unquote1(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))

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

(121) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(cons(X1, X2)) → ACTIVE(X1)
ACTIVE(unquote(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
s(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
from(x1)  =  x1
s1(x1)  =  x1
unquote(x1)  =  unquote(x1)
unquote1(x1)  =  x1

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

cons₂ > [ACTIVE₁, unquote₁]

Status:

cons₂: multiset
unquote₁: multiset
ACTIVE₁: multiset

The following usable rules [FROCOS05] were oriented: none

(122) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(s1(X)) → ACTIVE(X)
ACTIVE(unquote1(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))

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

(123) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(s1(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
s(x1)  =  x1
from(x1)  =  x1
s1(x1)  =  s1(x1)
unquote1(x1)  =  x1

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

[ACTIVE₁, s1₁]

Status:

s1₁: multiset
ACTIVE₁: multiset

The following usable rules [FROCOS05] were oriented: none

(124) Obligation:

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

ACTIVE(s(X)) → ACTIVE(X)
ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(unquote1(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))

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

(125) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(s(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
s(x1)  =  s(x1)
from(x1)  =  x1
unquote1(x1)  =  x1

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

[ACTIVE₁, s₁]

Status:

s₁: multiset
ACTIVE₁: multiset

The following usable rules [FROCOS05] were oriented: none

(126) Obligation:

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

ACTIVE(from(X)) → ACTIVE(X)
ACTIVE(unquote1(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))

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

(127) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(from(X)) → ACTIVE(X)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
ACTIVE(x1)  =  ACTIVE(x1)
from(x1)  =  from(x1)
unquote1(x1)  =  x1

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

[ACTIVE₁, from₁]

Status:

from₁: multiset
ACTIVE₁: multiset

The following usable rules [FROCOS05] were oriented: none

(128) Obligation:

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

ACTIVE(unquote1(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))

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

(129) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

ACTIVE(unquote1(X)) → ACTIVE(X)

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

unquote1₁ > ACTIVE₁

Status:

unquote1₁: multiset
ACTIVE₁: multiset

The following usable rules [FROCOS05] were oriented: none

(130) Obligation:

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

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

(131) PisEmptyProof (EQUIVALENT transformation)

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

(132) TRUE

(133) Obligation:

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

TOP(ok(X)) → TOP(active(X))
TOP(mark(X)) → TOP(proper(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.