(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: Combined order from the following AFS and order.
QUOTE1(x1)  =  QUOTE1(x1)
ok(x1)  =  ok(x1)
active(x1)  =  x1
sel(x1, x2)  =  x2
s(x1)  =  s(x1)
cons(x1, x2)  =  x1
mark(x1)  =  mark
0  =  0
first(x1, x2)  =  first(x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  x1
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1)
nil1  =  nil1
cons1(x1, x2)  =  x1
01  =  01
quote1(x1)  =  x1
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  unquote1(x1)
fcons(x1, x2)  =  fcons(x1)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
[ok1, s1, first1, first11, unquote11, fcons1, proper1] > QUOTE11 > [mark, nil]
[ok1, s1, first1, first11, unquote11, fcons1, proper1] > 0 > 01 > [mark, nil]
[ok1, s1, first1, first11, unquote11, fcons1, proper1] > nil1 > [mark, nil]
[ok1, s1, first1, first11, unquote11, fcons1, proper1] > top > [mark, nil]

Status:
QUOTE11: multiset
ok1: [1]
s1: [1]
mark: []
0: multiset
first1: [1]
nil: multiset
first11: [1]
nil1: multiset
01: multiset
unquote11: [1]
fcons1: [1]
proper1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(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: Combined order from the following AFS and order.
QUOTE(x1)  =  QUOTE(x1)
ok(x1)  =  ok(x1)
active(x1)  =  x1
sel(x1, x2)  =  x2
s(x1)  =  s(x1)
cons(x1, x2)  =  x1
mark(x1)  =  mark
0  =  0
first(x1, x2)  =  first(x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  x1
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1)
nil1  =  nil1
cons1(x1, x2)  =  x1
01  =  01
quote1(x1)  =  x1
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  unquote1(x1)
fcons(x1, x2)  =  fcons(x1)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
[ok1, s1, first1, first11, unquote11, fcons1, proper1] > QUOTE1 > [mark, nil]
[ok1, s1, first1, first11, unquote11, fcons1, proper1] > 0 > 01 > [mark, nil]
[ok1, s1, first1, first11, unquote11, fcons1, proper1] > nil1 > [mark, nil]
[ok1, s1, first1, first11, unquote11, fcons1, proper1] > top > [mark, nil]

Status:
QUOTE1: multiset
ok1: [1]
s1: [1]
mark: []
0: multiset
first1: [1]
nil: multiset
first11: [1]
nil1: multiset
01: multiset
unquote11: [1]
fcons1: [1]
proper1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

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

Recursive path order with status [RPO].
Quasi-Precedence:
active1 > cons2 > sel2 > [mark1, 0] > top
active1 > cons2 > first2 > [mark1, 0] > top
active1 > cons2 > sel12 > [mark1, 0] > top
active1 > cons2 > first12 > cons12 > fcons2 > [mark1, 0] > top
active1 > cons2 > quote11 > [mark1, 0] > top
active1 > [nil, nil1] > [mark1, 0] > top
active1 > 01 > [mark1, 0] > top

Status:
FCONS2: [2,1]
mark1: [1]
active1: [1]
sel2: [2,1]
cons2: multiset
0: multiset
first2: multiset
nil: multiset
sel12: [2,1]
first12: multiset
nil1: multiset
cons12: multiset
01: multiset
quote11: multiset
fcons2: [1,2]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

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

Recursive path order with status [RPO].
Quasi-Precedence:
[cons1, proper1] > [0, nil1] > [ok1, s1, quote1] > [mark, top]
[cons1, proper1] > [0, nil1] > nil > [mark, top]
[cons1, proper1] > unquote1 > [ok1, s1, quote1] > [mark, top]
01 > [0, nil1] > [ok1, s1, quote1] > [mark, top]
01 > [0, nil1] > nil > [mark, top]

Status:
ok1: multiset
s1: multiset
cons1: [1]
mark: multiset
0: multiset
nil: multiset
quote1: multiset
nil1: multiset
01: multiset
unquote1: multiset
proper1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

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

Recursive path order with status [RPO].
Quasi-Precedence:
UNQUOTE11 > [ok1, nil1]
top > active1 > first2 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > active1 > first2 > [cons2, fcons2] > first12 > [ok1, nil1]
top > active1 > first2 > nil > [ok1, nil1]
top > active1 > 01 > 0 > [ok1, nil1]
top > active1 > quote11 > cons12 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > active1 > quote11 > cons12 > [cons2, fcons2] > first12 > [ok1, nil1]
top > proper1 > first2 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > proper1 > first2 > [cons2, fcons2] > first12 > [ok1, nil1]
top > proper1 > first2 > nil > [ok1, nil1]
top > proper1 > 01 > 0 > [ok1, nil1]
top > proper1 > quote11 > cons12 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > proper1 > quote11 > cons12 > [cons2, fcons2] > first12 > [ok1, nil1]

Status:
UNQUOTE11: [1]
ok1: [1]
active1: [1]
cons2: multiset
0: multiset
first2: [2,1]
nil: multiset
sel11: [1]
first12: [2,1]
nil1: multiset
cons12: multiset
01: multiset
quote11: [1]
fcons2: [2,1]
proper1: multiset
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(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: Combined order from the following AFS and order.
UNQUOTE1(x1)  =  UNQUOTE1(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
sel(x1, x2)  =  sel(x1, x2)
s(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
first(x1, x2)  =  first(x1, x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x1, x2)
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1, x2)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x1, x2)
01  =  01
quote1(x1)  =  quote1(x1)
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1
fcons(x1, x2)  =  fcons(x1, x2)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
[active1, nil, nil1] > cons2 > sel2 > sel12 > [UNQUOTE11, mark1, top]
[active1, nil, nil1] > cons2 > first2 > [UNQUOTE11, mark1, top]
[active1, nil, nil1] > cons2 > quote11 > [UNQUOTE11, mark1, top]
[active1, nil, nil1] > [0, 01] > [UNQUOTE11, mark1, top]
[active1, nil, nil1] > first12 > cons12 > fcons2 > [UNQUOTE11, mark1, top]

Status:
UNQUOTE11: multiset
mark1: [1]
active1: [1]
sel2: [1,2]
cons2: [1,2]
0: multiset
first2: [1,2]
nil: multiset
sel12: [2,1]
first12: [2,1]
nil1: multiset
cons12: [2,1]
01: multiset
quote11: multiset
fcons2: [1,2]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

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

Recursive path order with status [RPO].
Quasi-Precedence:
UNQUOTE1 > [ok1, nil1]
top > active1 > first2 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > active1 > first2 > [cons2, fcons2] > first12 > [ok1, nil1]
top > active1 > first2 > nil > [ok1, nil1]
top > active1 > 01 > 0 > [ok1, nil1]
top > active1 > quote11 > cons12 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > active1 > quote11 > cons12 > [cons2, fcons2] > first12 > [ok1, nil1]
top > proper1 > first2 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > proper1 > first2 > [cons2, fcons2] > first12 > [ok1, nil1]
top > proper1 > first2 > nil > [ok1, nil1]
top > proper1 > 01 > 0 > [ok1, nil1]
top > proper1 > quote11 > cons12 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > proper1 > quote11 > cons12 > [cons2, fcons2] > first12 > [ok1, nil1]

Status:
UNQUOTE1: [1]
ok1: [1]
active1: [1]
cons2: multiset
0: multiset
first2: [2,1]
nil: multiset
sel11: [1]
first12: [2,1]
nil1: multiset
cons12: multiset
01: multiset
quote11: [1]
fcons2: [2,1]
proper1: multiset
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(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: Combined order from the following AFS and order.
UNQUOTE(x1)  =  UNQUOTE(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
sel(x1, x2)  =  sel(x1, x2)
s(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
first(x1, x2)  =  first(x1, x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x1, x2)
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1, x2)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x1, x2)
01  =  01
quote1(x1)  =  quote1(x1)
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1
fcons(x1, x2)  =  fcons(x1, x2)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
[active1, nil, nil1] > cons2 > sel2 > sel12 > [UNQUOTE1, mark1, top]
[active1, nil, nil1] > cons2 > first2 > [UNQUOTE1, mark1, top]
[active1, nil, nil1] > cons2 > quote11 > [UNQUOTE1, mark1, top]
[active1, nil, nil1] > [0, 01] > [UNQUOTE1, mark1, top]
[active1, nil, nil1] > first12 > cons12 > fcons2 > [UNQUOTE1, mark1, top]

Status:
UNQUOTE1: multiset
mark1: [1]
active1: [1]
sel2: [1,2]
cons2: [1,2]
0: multiset
first2: [1,2]
nil: multiset
sel12: [2,1]
first12: [2,1]
nil1: multiset
cons12: [2,1]
01: multiset
quote11: multiset
fcons2: [1,2]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

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

Recursive path order with status [RPO].
Quasi-Precedence:
S11 > [ok1, nil1]
top > active1 > first2 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > active1 > first2 > [cons2, fcons2] > first12 > [ok1, nil1]
top > active1 > first2 > nil > [ok1, nil1]
top > active1 > 01 > 0 > [ok1, nil1]
top > active1 > quote11 > cons12 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > active1 > quote11 > cons12 > [cons2, fcons2] > first12 > [ok1, nil1]
top > proper1 > first2 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > proper1 > first2 > [cons2, fcons2] > first12 > [ok1, nil1]
top > proper1 > first2 > nil > [ok1, nil1]
top > proper1 > 01 > 0 > [ok1, nil1]
top > proper1 > quote11 > cons12 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > proper1 > quote11 > cons12 > [cons2, fcons2] > first12 > [ok1, nil1]

Status:
S11: [1]
ok1: [1]
active1: [1]
cons2: multiset
0: multiset
first2: [2,1]
nil: multiset
sel11: [1]
first12: [2,1]
nil1: multiset
cons12: multiset
01: multiset
quote11: [1]
fcons2: [2,1]
proper1: multiset
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(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: Combined order from the following AFS and order.
S1(x1)  =  S1(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
sel(x1, x2)  =  sel(x1, x2)
s(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
first(x1, x2)  =  first(x1, x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x1, x2)
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1, x2)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x1, x2)
01  =  01
quote1(x1)  =  quote1(x1)
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1
fcons(x1, x2)  =  fcons(x1, x2)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
[active1, nil, nil1] > cons2 > sel2 > sel12 > [S11, mark1, top]
[active1, nil, nil1] > cons2 > first2 > [S11, mark1, top]
[active1, nil, nil1] > cons2 > quote11 > [S11, mark1, top]
[active1, nil, nil1] > [0, 01] > [S11, mark1, top]
[active1, nil, nil1] > first12 > cons12 > fcons2 > [S11, mark1, top]

Status:
S11: multiset
mark1: [1]
active1: [1]
sel2: [1,2]
cons2: [1,2]
0: multiset
first2: [1,2]
nil: multiset
sel12: [2,1]
first12: [2,1]
nil1: multiset
cons12: [2,1]
01: multiset
quote11: multiset
fcons2: [1,2]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

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

Recursive path order with status [RPO].
Quasi-Precedence:
active1 > cons2 > sel2 > [mark1, 0] > top
active1 > cons2 > first2 > [mark1, 0] > top
active1 > cons2 > sel12 > [mark1, 0] > top
active1 > cons2 > first12 > cons12 > fcons2 > [mark1, 0] > top
active1 > cons2 > quote11 > [mark1, 0] > top
active1 > [nil, nil1] > [mark1, 0] > top
active1 > 01 > [mark1, 0] > top

Status:
CONS12: [2,1]
mark1: [1]
active1: [1]
sel2: [2,1]
cons2: multiset
0: multiset
first2: multiset
nil: multiset
sel12: [2,1]
first12: multiset
nil1: multiset
cons12: multiset
01: multiset
quote11: multiset
fcons2: [1,2]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

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

Recursive path order with status [RPO].
Quasi-Precedence:
[cons1, proper1] > [0, nil1] > [ok1, s1, quote1] > [mark, top]
[cons1, proper1] > [0, nil1] > nil > [mark, top]
[cons1, proper1] > unquote1 > [ok1, s1, quote1] > [mark, top]
01 > [0, nil1] > [ok1, s1, quote1] > [mark, top]
01 > [0, nil1] > nil > [mark, top]

Status:
ok1: multiset
s1: multiset
cons1: [1]
mark: multiset
0: multiset
nil: multiset
quote1: multiset
nil1: multiset
01: multiset
unquote1: multiset
proper1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

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

Recursive path order with status [RPO].
Quasi-Precedence:
active1 > cons2 > sel2 > [mark1, 0] > top
active1 > cons2 > first2 > [mark1, 0] > top
active1 > cons2 > sel12 > [mark1, 0] > top
active1 > cons2 > first12 > cons12 > fcons2 > [mark1, 0] > top
active1 > cons2 > quote11 > [mark1, 0] > top
active1 > [nil, nil1] > [mark1, 0] > top
active1 > 01 > [mark1, 0] > top

Status:
FIRST12: [2,1]
mark1: [1]
active1: [1]
sel2: [2,1]
cons2: multiset
0: multiset
first2: multiset
nil: multiset
sel12: [2,1]
first12: multiset
nil1: multiset
cons12: multiset
01: multiset
quote11: multiset
fcons2: [1,2]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

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

Recursive path order with status [RPO].
Quasi-Precedence:
[cons1, proper1] > [0, nil1] > [ok1, s1, quote1] > [mark, top]
[cons1, proper1] > [0, nil1] > nil > [mark, top]
[cons1, proper1] > unquote1 > [ok1, s1, quote1] > [mark, top]
01 > [0, nil1] > [ok1, s1, quote1] > [mark, top]
01 > [0, nil1] > nil > [mark, top]

Status:
ok1: multiset
s1: multiset
cons1: [1]
mark: multiset
0: multiset
nil: multiset
quote1: multiset
nil1: multiset
01: multiset
unquote1: multiset
proper1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

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

Recursive path order with status [RPO].
Quasi-Precedence:
active1 > [0, 01]
active1 > from1 > mark1
active1 > [unquote11, fcons2] > cons2 > sel2 > sel12 > mark1
active1 > [unquote11, fcons2] > cons2 > first2 > mark1
active1 > [unquote11, fcons2] > cons2 > first12 > nil1 > mark1
active1 > [unquote11, fcons2] > cons2 > cons12 > mark1
active1 > [unquote11, fcons2] > nil > nil1 > mark1

Status:
mark1: [1]
active1: [1]
sel2: [2,1]
cons2: [1,2]
0: multiset
first2: [1,2]
nil: multiset
from1: multiset
sel12: [1,2]
first12: [2,1]
nil1: multiset
cons12: [2,1]
01: multiset
unquote11: [1]
fcons2: [1,2]
top: []


The following usable rules [FROCOS05] were oriented:

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))

(59) Obligation:

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

SEL1(X1, mark(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.

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

Recursive path order with status [RPO].
Quasi-Precedence:
SEL11 > [mark, top]
proper > 0 > ok1 > [mark, top]
proper > 0 > [nil, nil1] > [mark, top]
proper > 0 > 01 > [mark, top]

Status:
SEL11: [1]
mark: []
ok1: [1]
0: multiset
nil: multiset
nil1: multiset
01: multiset
proper: []
top: []


The following usable rules [FROCOS05] were oriented:

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))

(61) Obligation:

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

SEL1(X1, mark(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.

(62) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


SEL1(X1, mark(X2)) → SEL1(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
SEL1(x1, x2)  =  SEL1(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
sel(x1, x2)  =  sel(x1, x2)
s(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
first(x1, x2)  =  first(x1, x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x1, x2)
quote(x1)  =  quote(x1)
first1(x1, x2)  =  first1(x1, x2)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x1, x2)
01  =  01
quote1(x1)  =  quote1(x1)
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1
fcons(x1, x2)  =  fcons(x1, x2)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
active1 > first2 > cons2 > sel2 > mark1
active1 > first2 > cons2 > quote11 > mark1
active1 > first2 > nil > mark1
active1 > quote1 > sel12 > mark1
active1 > quote1 > 01 > mark1
active1 > quote1 > 01 > 0
active1 > first12 > mark1
active1 > nil1 > nil > mark1
active1 > [cons12, fcons2] > cons2 > sel2 > mark1
active1 > [cons12, fcons2] > cons2 > quote11 > mark1

Status:
SEL11: multiset
mark1: [1]
active1: [1]
sel2: [2,1]
cons2: [1,2]
0: multiset
first2: [1,2]
nil: multiset
sel12: [2,1]
quote1: [1]
first12: [2,1]
nil1: multiset
cons12: multiset
01: multiset
quote11: multiset
fcons2: multiset
top: []


The following usable rules [FROCOS05] were oriented:

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))

(63) Obligation:

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

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

(64) PisEmptyProof (EQUIVALENT transformation)

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

(65) TRUE

(66) Obligation:

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

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.

(67) 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
active(x1)  =  active(x1)
sel(x1, x2)  =  x2
s(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
first(x1, x2)  =  first(x1, x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x2)
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1, x2)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x1, x2)
01  =  01
quote1(x1)  =  quote1(x1)
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1
fcons(x1, x2)  =  fcons(x1, x2)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
FROM1 > [ok1, nil1]
top > active1 > first2 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > active1 > first2 > [cons2, fcons2] > first12 > [ok1, nil1]
top > active1 > first2 > nil > [ok1, nil1]
top > active1 > 01 > 0 > [ok1, nil1]
top > active1 > quote11 > cons12 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > active1 > quote11 > cons12 > [cons2, fcons2] > first12 > [ok1, nil1]
top > proper1 > first2 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > proper1 > first2 > [cons2, fcons2] > first12 > [ok1, nil1]
top > proper1 > first2 > nil > [ok1, nil1]
top > proper1 > 01 > 0 > [ok1, nil1]
top > proper1 > quote11 > cons12 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > proper1 > quote11 > cons12 > [cons2, fcons2] > first12 > [ok1, nil1]

Status:
FROM1: [1]
ok1: [1]
active1: [1]
cons2: multiset
0: multiset
first2: [2,1]
nil: multiset
sel11: [1]
first12: [2,1]
nil1: multiset
cons12: multiset
01: multiset
quote11: [1]
fcons2: [2,1]
proper1: multiset
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(68) 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.

(69) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive path order with status [RPO].
Quasi-Precedence:
[active1, nil, nil1] > cons2 > sel2 > sel12 > [FROM1, mark1, top]
[active1, nil, nil1] > cons2 > first2 > [FROM1, mark1, top]
[active1, nil, nil1] > cons2 > quote11 > [FROM1, mark1, top]
[active1, nil, nil1] > [0, 01] > [FROM1, mark1, top]
[active1, nil, nil1] > first12 > cons12 > fcons2 > [FROM1, mark1, top]

Status:
FROM1: multiset
mark1: [1]
active1: [1]
sel2: [1,2]
cons2: [1,2]
0: multiset
first2: [1,2]
nil: multiset
sel12: [2,1]
first12: [2,1]
nil1: multiset
cons12: [2,1]
01: multiset
quote11: multiset
fcons2: [1,2]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(70) Obligation:

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

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

(71) PisEmptyProof (EQUIVALENT transformation)

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

(72) TRUE

(73) Obligation:

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

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.

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

Recursive path order with status [RPO].
Quasi-Precedence:
active1 > [cons2, cons12] > first2 > nil > [FIRST1, ok1, from1]
active1 > [cons2, cons12] > first2 > nil > nil1
active1 > [cons2, cons12] > unquote1 > 0 > [FIRST1, ok1, from1]
active1 > [cons2, cons12] > unquote1 > 0 > nil1
active1 > [cons2, cons12] > unquote1 > 0 > 01
active1 > [cons2, cons12] > fcons2 > [FIRST1, ok1, from1]
proper1 > [cons2, cons12] > first2 > nil > [FIRST1, ok1, from1]
proper1 > [cons2, cons12] > first2 > nil > nil1
proper1 > [cons2, cons12] > unquote1 > 0 > [FIRST1, ok1, from1]
proper1 > [cons2, cons12] > unquote1 > 0 > nil1
proper1 > [cons2, cons12] > unquote1 > 0 > 01
proper1 > [cons2, cons12] > fcons2 > [FIRST1, ok1, from1]

Status:
FIRST1: multiset
ok1: [1]
active1: [1]
cons2: multiset
0: multiset
first2: multiset
nil: multiset
from1: [1]
nil1: multiset
cons12: [2,1]
01: multiset
unquote1: multiset
fcons2: [2,1]
proper1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(75) Obligation:

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

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

The TRS R consists of the following rules:

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

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


FIRST(mark(X1), X2) → FIRST(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FIRST(x1, x2)  =  FIRST(x1)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
sel(x1, x2)  =  sel(x1, x2)
s(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
first(x1, x2)  =  first(x1, x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x1, x2)
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1, x2)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x1, x2)
01  =  01
quote1(x1)  =  x1
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1
fcons(x1, x2)  =  fcons(x1, x2)
proper(x1)  =  proper(x1)
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
FIRST1 > mark1
top > proper1 > [active1, sel12, nil1] > first2 > [sel2, cons2, fcons2] > cons12 > mark1
top > proper1 > [active1, sel12, nil1] > first2 > nil > mark1
top > proper1 > [active1, sel12, nil1] > first2 > first12 > cons12 > mark1
top > proper1 > [active1, sel12, nil1] > 01 > 0 > mark1

Status:
FIRST1: multiset
mark1: [1]
active1: multiset
sel2: multiset
cons2: [2,1]
0: multiset
first2: [2,1]
nil: multiset
sel12: multiset
first12: multiset
nil1: multiset
cons12: [1,2]
01: multiset
fcons2: [1,2]
proper1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(77) Obligation:

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

FIRST(X1, mark(X2)) → FIRST(X1, X2)

The TRS R consists of the following rules:

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

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

(78) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


FIRST(X1, mark(X2)) → FIRST(X1, X2)
The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
FIRST(x1, x2)  =  FIRST(x2)
mark(x1)  =  mark(x1)
active(x1)  =  active(x1)
sel(x1, x2)  =  sel(x1, x2)
s(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
first(x1, x2)  =  first(x1, x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x1, x2)
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1, x2)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x1, x2)
01  =  01
quote1(x1)  =  x1
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  unquote1(x1)
fcons(x1, x2)  =  fcons(x1, x2)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
FIRST1 > top
[active1, unquote11] > cons2 > sel2 > mark1 > top
[active1, unquote11] > cons2 > first2 > mark1 > top
[active1, unquote11] > cons2 > first2 > [nil, nil1] > top
[active1, unquote11] > cons2 > sel12 > mark1 > top
[active1, unquote11] > cons2 > first12 > [nil, nil1] > top
[active1, unquote11] > cons2 > first12 > cons12 > mark1 > top
[active1, unquote11] > 01 > mark1 > top
[active1, unquote11] > 01 > 0 > [nil, nil1] > top
[active1, unquote11] > fcons2 > mark1 > top

Status:
FIRST1: multiset
mark1: [1]
active1: [1]
sel2: [2,1]
cons2: [1,2]
0: multiset
first2: multiset
nil: multiset
sel12: [1,2]
first12: [1,2]
nil1: multiset
cons12: multiset
01: multiset
unquote11: [1]
fcons2: multiset
top: []


The following usable rules [FROCOS05] were oriented:

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))

(79) 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.

(80) PisEmptyProof (EQUIVALENT transformation)

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

(81) TRUE

(82) 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.

(83) 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
active(x1)  =  x1
sel(x1, x2)  =  sel(x2)
s(x1)  =  s(x1)
cons(x1, x2)  =  x1
0  =  0
first(x1, x2)  =  x1
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x1)
quote(x1)  =  x1
first1(x1, x2)  =  x1
nil1  =  nil1
cons1(x1, x2)  =  x1
01  =  01
quote1(x1)  =  x1
s1(x1)  =  s1(x1)
unquote(x1)  =  unquote(x1)
unquote1(x1)  =  unquote1(x1)
fcons(x1, x2)  =  x1
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
CONS1 > [ok1, mark]
top > proper1 > s1 > sel1 > [ok1, mark]
top > proper1 > s1 > s11 > [0, nil1, unquote1] > [ok1, mark]
top > proper1 > sel11 > [ok1, mark]
top > proper1 > 01 > [0, nil1, unquote1] > [ok1, mark]
top > proper1 > unquote11 > nil > [0, nil1, unquote1] > [ok1, mark]

Status:
CONS1: multiset
ok1: [1]
mark: []
sel1: [1]
s1: [1]
0: multiset
nil: multiset
sel11: [1]
nil1: multiset
01: multiset
s11: [1]
unquote1: [1]
unquote11: multiset
proper1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(84) 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.

(85) 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)
active(x1)  =  active(x1)
sel(x1, x2)  =  sel(x1, x2)
s(x1)  =  s(x1)
cons(x1, x2)  =  cons(x1, x2)
0  =  0
first(x1, x2)  =  first(x1, x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x1, x2)
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1, x2)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x1, x2)
01  =  01
quote1(x1)  =  x1
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1
fcons(x1, x2)  =  fcons(x1, x2)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
[active1, sel12, top] > cons2 > sel2 > [mark1, s1]
[active1, sel12, top] > cons2 > [first2, first12, nil1] > nil > [mark1, s1]
[active1, sel12, top] > cons12 > fcons2 > [mark1, s1]
[active1, sel12, top] > 01 > [mark1, s1]
[active1, sel12, top] > 01 > 0

Status:
CONS1: multiset
mark1: [1]
active1: [1]
sel2: [1,2]
s1: [1]
cons2: [1,2]
0: multiset
first2: multiset
nil: multiset
sel12: [2,1]
first12: [2,1]
nil1: multiset
cons12: [1,2]
01: multiset
fcons2: multiset
top: []


The following usable rules [FROCOS05] were oriented:

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))

(86) 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.

(87) PisEmptyProof (EQUIVALENT transformation)

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

(88) TRUE

(89) 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.

(90) 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
active(x1)  =  active(x1)
sel(x1, x2)  =  x2
s(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
first(x1, x2)  =  first(x1, x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x2)
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1, x2)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x1, x2)
01  =  01
quote1(x1)  =  quote1(x1)
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1
fcons(x1, x2)  =  fcons(x1, x2)
proper(x1)  =  proper(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
S1 > [ok1, nil1]
top > active1 > first2 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > active1 > first2 > [cons2, fcons2] > first12 > [ok1, nil1]
top > active1 > first2 > nil > [ok1, nil1]
top > active1 > 01 > 0 > [ok1, nil1]
top > active1 > quote11 > cons12 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > active1 > quote11 > cons12 > [cons2, fcons2] > first12 > [ok1, nil1]
top > proper1 > first2 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > proper1 > first2 > [cons2, fcons2] > first12 > [ok1, nil1]
top > proper1 > first2 > nil > [ok1, nil1]
top > proper1 > 01 > 0 > [ok1, nil1]
top > proper1 > quote11 > cons12 > [cons2, fcons2] > sel11 > [ok1, nil1]
top > proper1 > quote11 > cons12 > [cons2, fcons2] > first12 > [ok1, nil1]

Status:
S1: [1]
ok1: [1]
active1: [1]
cons2: multiset
0: multiset
first2: [2,1]
nil: multiset
sel11: [1]
first12: [2,1]
nil1: multiset
cons12: multiset
01: multiset
quote11: [1]
fcons2: [2,1]
proper1: multiset
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(91) 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.

(92) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


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

Recursive path order with status [RPO].
Quasi-Precedence:
[active1, nil, nil1] > cons2 > sel2 > sel12 > [S1, mark1, top]
[active1, nil, nil1] > cons2 > first2 > [S1, mark1, top]
[active1, nil, nil1] > cons2 > quote11 > [S1, mark1, top]
[active1, nil, nil1] > [0, 01] > [S1, mark1, top]
[active1, nil, nil1] > first12 > cons12 > fcons2 > [S1, mark1, top]

Status:
S1: multiset
mark1: [1]
active1: [1]
sel2: [1,2]
cons2: [1,2]
0: multiset
first2: [1,2]
nil: multiset
sel12: [2,1]
first12: [2,1]
nil1: multiset
cons12: [2,1]
01: multiset
quote11: multiset
fcons2: [1,2]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(93) 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.

(94) PisEmptyProof (EQUIVALENT transformation)

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

(95) TRUE

(96) 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.

(97) 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
active(x1)  =  active(x1)
sel(x1, x2)  =  sel(x1, x2)
s(x1)  =  x1
cons(x1, x2)  =  cons(x1, x2)
0  =  0
first(x1, x2)  =  first(x1, x2)
nil  =  nil
from(x1)  =  x1
sel1(x1, x2)  =  sel1(x1, x2)
quote(x1)  =  x1
first1(x1, x2)  =  first1(x1, x2)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x1, x2)
01  =  01
quote1(x1)  =  quote1(x1)
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1
fcons(x1, x2)  =  fcons(x1, x2)
proper(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
active1 > cons2 > sel2 > [mark1, 0] > top
active1 > cons2 > first2 > [mark1, 0] > top
active1 > cons2 > sel12 > [mark1, 0] > top
active1 > cons2 > first12 > cons12 > fcons2 > [mark1, 0] > top
active1 > cons2 > quote11 > [mark1, 0] > top
active1 > [nil, nil1] > [mark1, 0] > top
active1 > 01 > [mark1, 0] > top

Status:
SEL2: [2,1]
mark1: [1]
active1: [1]
sel2: [2,1]
cons2: multiset
0: multiset
first2: multiset
nil: multiset
sel12: [2,1]
first12: multiset
nil1: multiset
cons12: multiset
01: multiset
quote11: multiset
fcons2: [1,2]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(98) 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.

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

Recursive path order with status [RPO].
Quasi-Precedence:
[cons1, proper1] > [0, nil1] > [ok1, s1, quote1] > [mark, top]
[cons1, proper1] > [0, nil1] > nil > [mark, top]
[cons1, proper1] > unquote1 > [ok1, s1, quote1] > [mark, top]
01 > [0, nil1] > [ok1, s1, quote1] > [mark, top]
01 > [0, nil1] > nil > [mark, top]

Status:
ok1: multiset
s1: multiset
cons1: [1]
mark: multiset
0: multiset
nil: multiset
quote1: multiset
nil1: multiset
01: multiset
unquote1: multiset
proper1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(100) 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.

(101) PisEmptyProof (EQUIVALENT transformation)

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

(102) TRUE

(103) 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.

(104) 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(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(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 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)  =  quote(x1)
first1(x1, x2)  =  first1(x1, x2)
cons1(x1, x2)  =  cons1(x1, x2)
quote1(x1)  =  x1
s1(x1)  =  s1(x1)
unquote(x1)  =  unquote(x1)
unquote1(x1)  =  unquote1(x1)
fcons(x1, x2)  =  fcons(x1, x2)
active(x1)  =  x1
mark(x1)  =  mark
0  =  0
nil  =  nil
nil1  =  nil1
01  =  01
proper(x1)  =  proper(x1)
ok(x1)  =  ok(x1)
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
proper1 > cons2 > sel2 > PROPER1 > [mark, ok1]
proper1 > cons2 > sel2 > sel12 > [mark, ok1]
proper1 > cons2 > [first2, quote1, first12] > sel12 > [mark, ok1]
proper1 > cons2 > [first2, quote1, first12] > nil1 > [mark, ok1]
proper1 > cons2 > cons12 > PROPER1 > [mark, ok1]
proper1 > s11 > PROPER1 > [mark, ok1]
proper1 > unquote11 > unquote1 > PROPER1 > [mark, ok1]
proper1 > unquote11 > unquote1 > 0 > [first2, quote1, first12] > sel12 > [mark, ok1]
proper1 > unquote11 > unquote1 > 0 > [first2, quote1, first12] > nil1 > [mark, ok1]
proper1 > unquote11 > fcons2 > PROPER1 > [mark, ok1]
proper1 > unquote11 > nil > nil1 > [mark, ok1]
proper1 > 01 > 0 > [first2, quote1, first12] > sel12 > [mark, ok1]
proper1 > 01 > 0 > [first2, quote1, first12] > nil1 > [mark, ok1]
top > [mark, ok1]

Status:
PROPER1: [1]
sel2: [2,1]
cons2: [1,2]
first2: multiset
sel12: [2,1]
quote1: multiset
first12: multiset
cons12: [2,1]
s11: [1]
unquote1: [1]
unquote11: multiset
fcons2: [2,1]
mark: []
0: multiset
nil: multiset
nil1: multiset
01: multiset
proper1: [1]
ok1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(105) Obligation:

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

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

Recursive path order with status [RPO].
Quasi-Precedence:
quote11 > PROPER1 > [mark, first1, top]
quote11 > nil1 > [mark, first1, top]
quote11 > cons11 > [mark, first1, top]
cons > sel1 > [mark, first1, top]
cons > cons11 > [mark, first1, top]
nil > nil1 > [mark, first1, top]
01 > 0 > [mark, first1, top]
s1 > [mark, first1, top]
unquote > 0 > [mark, first1, top]

Status:
PROPER1: multiset
quote11: [1]
cons: []
mark: []
0: multiset
nil: multiset
sel1: multiset
first1: []
nil1: multiset
cons11: [1]
01: multiset
s1: []
unquote: []
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(107) Obligation:

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

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

Recursive path order with status [RPO].
Quasi-Precedence:
PROPER1 > [mark, quote1, ok]
nil > nil1 > [mark, quote1, ok]
01 > 0 > [mark, quote1, ok]
[proper1, top] > quote1 > sel1 > active1 > s1 > first12 > [mark, quote1, ok]
[proper1, top] > quote1 > sel1 > active1 > sel1 > [mark, quote1, ok]
[proper1, top] > quote1 > sel1 > active1 > 0 > [mark, quote1, ok]
[proper1, top] > nil1 > [mark, quote1, ok]
[proper1, top] > cons1 > active1 > s1 > first12 > [mark, quote1, ok]
[proper1, top] > cons1 > active1 > sel1 > [mark, quote1, ok]
[proper1, top] > cons1 > active1 > 0 > [mark, quote1, ok]
[proper1, top] > unquote > active1 > s1 > first12 > [mark, quote1, ok]
[proper1, top] > unquote > active1 > sel1 > [mark, quote1, ok]
[proper1, top] > unquote > active1 > 0 > [mark, quote1, ok]
[proper1, top] > fcons > active1 > s1 > first12 > [mark, quote1, ok]
[proper1, top] > fcons > active1 > sel1 > [mark, quote1, ok]
[proper1, top] > fcons > active1 > 0 > [mark, quote1, ok]

Status:
PROPER1: multiset
s1: multiset
active1: [1]
sel1: multiset
mark: []
0: multiset
nil: multiset
sel1: []
quote1: multiset
first12: [2,1]
nil1: multiset
cons1: multiset
01: multiset
quote1: []
unquote: []
fcons: []
proper1: [1]
ok: []
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(109) Obligation:

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

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

Recursive path order with status [RPO].
Quasi-Precedence:
PROPER1 > [mark, sel12, ok]
from1 > [mark, sel12, ok]
01 > [0, nil, nil1] > [mark, sel12, ok]
quote11 > [0, nil, nil1] > [mark, sel12, ok]
quote11 > cons1 > [mark, sel12, ok]
unquote1 > [0, nil, nil1] > [mark, sel12, ok]
fcons2 > [mark, sel12, ok]
top > [mark, sel12, ok]

Status:
PROPER1: multiset
from1: multiset
mark: []
0: multiset
nil: multiset
sel12: multiset
nil1: multiset
cons1: multiset
01: multiset
quote11: multiset
unquote1: multiset
fcons2: [2,1]
ok: []
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(111) 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.

(112) PisEmptyProof (EQUIVALENT transformation)

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

(113) TRUE

(114) 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.

(115) 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)
active(x1)  =  x1
mark(x1)  =  mark
0  =  0
nil  =  nil
quote(x1)  =  quote
nil1  =  nil1
01  =  01
quote1(x1)  =  quote1(x1)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
sel2 > sel12 > quote > [ACTIVE1, mark]
first2 > nil > [first12, nil1] > [ACTIVE1, mark]
cons12 > [ACTIVE1, mark]
fcons2 > [ACTIVE1, mark]
0 > nil > [first12, nil1] > [ACTIVE1, mark]
0 > quote > [ACTIVE1, mark]
0 > 01 > [ACTIVE1, mark]
quote11 > [first12, nil1] > [ACTIVE1, mark]
top > [ACTIVE1, mark]

Status:
ACTIVE1: multiset
sel2: [1,2]
first2: [2,1]
sel12: multiset
first12: [2,1]
cons12: [2,1]
fcons2: [1,2]
mark: []
0: multiset
nil: multiset
quote: []
nil1: multiset
01: multiset
quote11: [1]
top: []


The following usable rules [FROCOS05] were oriented:

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))

(116) 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.

(117) 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)
ACTIVE(cons(X1, X2)) → ACTIVE(X1)
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)
cons(x1, x2)  =  cons(x1, x2)
from(x1)  =  x1
s1(x1)  =  x1
unquote(x1)  =  x1
unquote1(x1)  =  x1
active(x1)  =  x1
sel(x1, x2)  =  sel
mark(x1)  =  mark
0  =  0
first(x1, x2)  =  x2
nil  =  nil
sel1(x1, x2)  =  sel1(x1, x2)
quote(x1)  =  quote(x1)
first1(x1, x2)  =  first1(x1)
nil1  =  nil1
cons1(x1, x2)  =  cons1(x2)
01  =  01
quote1(x1)  =  x1
fcons(x1, x2)  =  fcons(x1)
proper(x1)  =  x1
ok(x1)  =  x1
top(x1)  =  top

Recursive path order with status [RPO].
Quasi-Precedence:
ACTIVE1 > [mark, top]
s1 > [cons2, first11, cons11] > sel > [mark, top]
s1 > [cons2, first11, cons11] > sel12 > [quote1, 01] > [mark, top]
s1 > [cons2, first11, cons11] > nil1 > nil > [mark, top]
0 > [quote1, 01] > [mark, top]
fcons1 > [cons2, first11, cons11] > sel > [mark, top]
fcons1 > [cons2, first11, cons11] > sel12 > [quote1, 01] > [mark, top]
fcons1 > [cons2, first11, cons11] > nil1 > nil > [mark, top]

Status:
ACTIVE1: [1]
s1: [1]
cons2: [1,2]
sel: []
mark: []
0: multiset
nil: multiset
sel12: [1,2]
quote1: [1]
first11: [1]
nil1: multiset
cons11: [1]
01: multiset
fcons1: [1]
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(118) Obligation:

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

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.

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

Recursive path order with status [RPO].
Quasi-Precedence:
[0, 01] > quote > s11 > s1 > sel2 > [cons, mark, nil, nil1] > ok
[0, 01] > quote > s11 > s1 > first11 > [cons, mark, nil, nil1] > ok
[0, 01] > quote > sel11 > [cons, mark, nil, nil1] > ok
first > active1 > quote > s11 > s1 > sel2 > [cons, mark, nil, nil1] > ok
first > active1 > quote > s11 > s1 > first11 > [cons, mark, nil, nil1] > ok
first > active1 > quote > sel11 > [cons, mark, nil, nil1] > ok
first > active1 > cons11 > fcons1 > [cons, mark, nil, nil1] > ok

Status:
s11: [1]
active1: [1]
sel2: multiset
s1: [1]
cons: []
mark: []
0: multiset
first: []
nil: multiset
sel11: [1]
quote: []
first11: multiset
nil1: multiset
cons11: multiset
01: multiset
fcons1: multiset
ok: []
top: []


The following usable rules [FROCOS05] were oriented:

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))

(120) Obligation:

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

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

Recursive path order with status [RPO].
Quasi-Precedence:
ACTIVE1 > mark
0 > nil1 > [unquote1, cons1, ok1, top] > mark
0 > nil1 > nil > mark
0 > 01 > [unquote1, cons1, ok1, top] > mark
proper1 > [unquote11, fcons2] > [unquote1, cons1, ok1, top] > mark
proper1 > [unquote11, fcons2] > nil > mark
proper1 > first12 > cons11 > [unquote1, cons1, ok1, top] > mark

Status:
ACTIVE1: multiset
unquote1: [1]
unquote11: [1]
cons1: [1]
mark: []
0: multiset
nil: multiset
first12: multiset
nil1: multiset
cons11: multiset
01: multiset
fcons2: multiset
proper1: [1]
ok1: [1]
top: []


The following usable rules [FROCOS05] were oriented:

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))

(122) Obligation:

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

ACTIVE(from(X)) → ACTIVE(X)

The TRS R consists of the following rules:

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

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

Recursive path order with status [RPO].
Quasi-Precedence:
ACTIVE1 > [mark, sel12, ok]
from1 > [mark, sel12, ok]
01 > [0, nil, nil1] > [mark, sel12, ok]
quote11 > [0, nil, nil1] > [mark, sel12, ok]
quote11 > cons1 > [mark, sel12, ok]
unquote1 > [0, nil, nil1] > [mark, sel12, ok]
fcons2 > [mark, sel12, ok]
top > [mark, sel12, ok]

Status:
ACTIVE1: multiset
from1: multiset
mark: []
0: multiset
nil: multiset
sel12: multiset
nil1: multiset
cons1: multiset
01: multiset
quote11: multiset
unquote1: multiset
fcons2: [2,1]
ok: []
top: multiset


The following usable rules [FROCOS05] were oriented:

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))

(124) 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.

(125) PisEmptyProof (EQUIVALENT transformation)

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

(126) TRUE

(127) 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.