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

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
PROPER1(from1(X)) -> FROM1(proper1(X))
ACTIVE1(quote11(cons2(X, Z))) -> CONS12(quote1(X), quote11(Z))
ACTIVE1(sel12(s1(X), cons2(Y, Z))) -> SEL12(X, Z)
ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(from1(X)) -> FROM1(active1(X))
PROPER1(quote11(X)) -> QUOTE11(proper1(X))
PROPER1(quote11(X)) -> PROPER1(X)
S1(mark1(X)) -> S1(X)
PROPER1(fcons2(X1, X2)) -> FCONS2(proper1(X1), proper1(X2))
ACTIVE1(sel12(0, cons2(X, Z))) -> QUOTE1(X)
QUOTE11(ok1(X)) -> QUOTE11(X)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote1(s11(X))) -> S1(unquote1(X))
TOP1(mark1(X)) -> TOP1(proper1(X))
ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons12(X1, X2)) -> CONS12(X1, active1(X2))
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(first12(X1, X2)) -> FIRST12(active1(X1), X2)
PROPER1(unquote11(X)) -> UNQUOTE11(proper1(X))
UNQUOTE1(ok1(X)) -> UNQUOTE1(X)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
PROPER1(sel12(X1, X2)) -> SEL12(proper1(X1), proper1(X2))
ACTIVE1(quote1(s1(X))) -> S11(quote1(X))
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
FIRST12(ok1(X1), ok1(X2)) -> FIRST12(X1, X2)
ACTIVE1(sel12(X1, X2)) -> SEL12(active1(X1), X2)
CONS12(X1, mark1(X2)) -> CONS12(X1, X2)
PROPER1(cons12(X1, X2)) -> PROPER1(X2)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
S11(ok1(X)) -> S11(X)
ACTIVE1(fcons2(X, Z)) -> CONS2(X, Z)
ACTIVE1(first2(X1, X2)) -> FIRST2(X1, active1(X2))
ACTIVE1(sel2(X1, X2)) -> SEL2(X1, active1(X2))
ACTIVE1(unquote11(X)) -> UNQUOTE11(active1(X))
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
QUOTE1(ok1(X)) -> QUOTE1(X)
S11(mark1(X)) -> S11(X)
ACTIVE1(first12(X1, X2)) -> FIRST12(X1, active1(X2))
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
PROPER1(quote1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
ACTIVE1(unquote11(cons12(X, Z))) -> UNQUOTE11(Z)
ACTIVE1(unquote11(cons12(X, Z))) -> UNQUOTE1(X)
PROPER1(first2(X1, X2)) -> FIRST2(proper1(X1), proper1(X2))
ACTIVE1(first12(s1(X), cons2(Y, Z))) -> QUOTE1(Y)
ACTIVE1(s11(X)) -> S11(active1(X))
FIRST2(ok1(X1), ok1(X2)) -> FIRST2(X1, X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
TOP1(ok1(X)) -> TOP1(active1(X))
FCONS2(ok1(X1), ok1(X2)) -> FCONS2(X1, X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> FROM1(s1(X))
FIRST2(X1, mark1(X2)) -> FIRST2(X1, X2)
ACTIVE1(quote1(sel2(X, Z))) -> SEL12(X, Z)
PROPER1(unquote11(X)) -> PROPER1(X)
ACTIVE1(unquote11(cons12(X, Z))) -> FCONS2(unquote1(X), unquote11(Z))
SEL12(mark1(X1), X2) -> SEL12(X1, X2)
PROPER1(unquote1(X)) -> UNQUOTE1(proper1(X))
PROPER1(s11(X)) -> PROPER1(X)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)
UNQUOTE1(mark1(X)) -> UNQUOTE1(X)
FIRST12(mark1(X1), X2) -> FIRST12(X1, X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
PROPER1(quote1(X)) -> QUOTE1(proper1(X))
ACTIVE1(first2(X1, X2)) -> FIRST2(active1(X1), X2)
ACTIVE1(quote1(s1(X))) -> QUOTE1(X)
CONS12(mark1(X1), X2) -> CONS12(X1, X2)
ACTIVE1(unquote1(s11(X))) -> UNQUOTE1(X)
ACTIVE1(unquote1(X)) -> UNQUOTE1(active1(X))
FROM1(mark1(X)) -> FROM1(X)
FIRST12(X1, mark1(X2)) -> FIRST12(X1, X2)
ACTIVE1(quote11(cons2(X, Z))) -> QUOTE11(Z)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(quote11(first2(X, Z))) -> FIRST12(X, Z)
PROPER1(s11(X)) -> S11(proper1(X))
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
TOP1(ok1(X)) -> ACTIVE1(X)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
ACTIVE1(s1(X)) -> S1(active1(X))
UNQUOTE11(mark1(X)) -> UNQUOTE11(X)
ACTIVE1(first12(s1(X), cons2(Y, Z))) -> FIRST12(X, Z)
FCONS2(mark1(X1), X2) -> FCONS2(X1, X2)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(first2(s1(X), cons2(Y, Z))) -> FIRST2(X, Z)
TOP1(mark1(X)) -> PROPER1(X)
SEL12(X1, mark1(X2)) -> SEL12(X1, X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
ACTIVE1(quote11(cons2(X, Z))) -> QUOTE1(X)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(sel2(X1, X2)) -> SEL2(proper1(X1), proper1(X2))
PROPER1(first2(X1, X2)) -> PROPER1(X2)
ACTIVE1(fcons2(X1, X2)) -> FCONS2(active1(X1), X2)
PROPER1(first12(X1, X2)) -> FIRST12(proper1(X1), proper1(X2))
UNQUOTE11(ok1(X)) -> UNQUOTE11(X)
ACTIVE1(sel12(X1, X2)) -> SEL12(X1, active1(X2))
ACTIVE1(sel2(s1(X), cons2(Y, Z))) -> SEL2(X, Z)
PROPER1(s1(X)) -> S1(proper1(X))
CONS12(ok1(X1), ok1(X2)) -> CONS12(X1, X2)
ACTIVE1(fcons2(X1, X2)) -> FCONS2(X1, active1(X2))
S1(ok1(X)) -> S1(X)
ACTIVE1(first2(s1(X), cons2(Y, Z))) -> CONS2(Y, first2(X, Z))
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons12(X1, X2)) -> CONS12(active1(X1), X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(first12(s1(X), cons2(Y, Z))) -> CONS12(quote1(Y), first12(X, Z))
PROPER1(cons12(X1, X2)) -> CONS12(proper1(X1), proper1(X2))
SEL12(ok1(X1), ok1(X2)) -> SEL12(X1, X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
ACTIVE1(from1(X)) -> S1(X)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
PROPER1(from1(X)) -> PROPER1(X)
ACTIVE1(sel2(X1, X2)) -> SEL2(active1(X1), X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
FCONS2(X1, mark1(X2)) -> FCONS2(X1, X2)
FROM1(ok1(X)) -> FROM1(X)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X1)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ DependencyGraphProof

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

ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
PROPER1(from1(X)) -> FROM1(proper1(X))
ACTIVE1(quote11(cons2(X, Z))) -> CONS12(quote1(X), quote11(Z))
ACTIVE1(sel12(s1(X), cons2(Y, Z))) -> SEL12(X, Z)
ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(from1(X)) -> FROM1(active1(X))
PROPER1(quote11(X)) -> QUOTE11(proper1(X))
PROPER1(quote11(X)) -> PROPER1(X)
S1(mark1(X)) -> S1(X)
PROPER1(fcons2(X1, X2)) -> FCONS2(proper1(X1), proper1(X2))
ACTIVE1(sel12(0, cons2(X, Z))) -> QUOTE1(X)
QUOTE11(ok1(X)) -> QUOTE11(X)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote1(s11(X))) -> S1(unquote1(X))
TOP1(mark1(X)) -> TOP1(proper1(X))
ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons12(X1, X2)) -> CONS12(X1, active1(X2))
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(first12(X1, X2)) -> FIRST12(active1(X1), X2)
PROPER1(unquote11(X)) -> UNQUOTE11(proper1(X))
UNQUOTE1(ok1(X)) -> UNQUOTE1(X)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
PROPER1(sel12(X1, X2)) -> SEL12(proper1(X1), proper1(X2))
ACTIVE1(quote1(s1(X))) -> S11(quote1(X))
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
FIRST12(ok1(X1), ok1(X2)) -> FIRST12(X1, X2)
ACTIVE1(sel12(X1, X2)) -> SEL12(active1(X1), X2)
CONS12(X1, mark1(X2)) -> CONS12(X1, X2)
PROPER1(cons12(X1, X2)) -> PROPER1(X2)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
S11(ok1(X)) -> S11(X)
ACTIVE1(fcons2(X, Z)) -> CONS2(X, Z)
ACTIVE1(first2(X1, X2)) -> FIRST2(X1, active1(X2))
ACTIVE1(sel2(X1, X2)) -> SEL2(X1, active1(X2))
ACTIVE1(unquote11(X)) -> UNQUOTE11(active1(X))
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
QUOTE1(ok1(X)) -> QUOTE1(X)
S11(mark1(X)) -> S11(X)
ACTIVE1(first12(X1, X2)) -> FIRST12(X1, active1(X2))
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
PROPER1(quote1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
ACTIVE1(unquote11(cons12(X, Z))) -> UNQUOTE11(Z)
ACTIVE1(unquote11(cons12(X, Z))) -> UNQUOTE1(X)
PROPER1(first2(X1, X2)) -> FIRST2(proper1(X1), proper1(X2))
ACTIVE1(first12(s1(X), cons2(Y, Z))) -> QUOTE1(Y)
ACTIVE1(s11(X)) -> S11(active1(X))
FIRST2(ok1(X1), ok1(X2)) -> FIRST2(X1, X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
TOP1(ok1(X)) -> TOP1(active1(X))
FCONS2(ok1(X1), ok1(X2)) -> FCONS2(X1, X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> FROM1(s1(X))
FIRST2(X1, mark1(X2)) -> FIRST2(X1, X2)
ACTIVE1(quote1(sel2(X, Z))) -> SEL12(X, Z)
PROPER1(unquote11(X)) -> PROPER1(X)
ACTIVE1(unquote11(cons12(X, Z))) -> FCONS2(unquote1(X), unquote11(Z))
SEL12(mark1(X1), X2) -> SEL12(X1, X2)
PROPER1(unquote1(X)) -> UNQUOTE1(proper1(X))
PROPER1(s11(X)) -> PROPER1(X)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)
UNQUOTE1(mark1(X)) -> UNQUOTE1(X)
FIRST12(mark1(X1), X2) -> FIRST12(X1, X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
PROPER1(quote1(X)) -> QUOTE1(proper1(X))
ACTIVE1(first2(X1, X2)) -> FIRST2(active1(X1), X2)
ACTIVE1(quote1(s1(X))) -> QUOTE1(X)
CONS12(mark1(X1), X2) -> CONS12(X1, X2)
ACTIVE1(unquote1(s11(X))) -> UNQUOTE1(X)
ACTIVE1(unquote1(X)) -> UNQUOTE1(active1(X))
FROM1(mark1(X)) -> FROM1(X)
FIRST12(X1, mark1(X2)) -> FIRST12(X1, X2)
ACTIVE1(quote11(cons2(X, Z))) -> QUOTE11(Z)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(quote11(first2(X, Z))) -> FIRST12(X, Z)
PROPER1(s11(X)) -> S11(proper1(X))
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
TOP1(ok1(X)) -> ACTIVE1(X)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
ACTIVE1(s1(X)) -> S1(active1(X))
UNQUOTE11(mark1(X)) -> UNQUOTE11(X)
ACTIVE1(first12(s1(X), cons2(Y, Z))) -> FIRST12(X, Z)
FCONS2(mark1(X1), X2) -> FCONS2(X1, X2)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
ACTIVE1(first2(s1(X), cons2(Y, Z))) -> FIRST2(X, Z)
TOP1(mark1(X)) -> PROPER1(X)
SEL12(X1, mark1(X2)) -> SEL12(X1, X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
ACTIVE1(quote11(cons2(X, Z))) -> QUOTE1(X)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(sel2(X1, X2)) -> SEL2(proper1(X1), proper1(X2))
PROPER1(first2(X1, X2)) -> PROPER1(X2)
ACTIVE1(fcons2(X1, X2)) -> FCONS2(active1(X1), X2)
PROPER1(first12(X1, X2)) -> FIRST12(proper1(X1), proper1(X2))
UNQUOTE11(ok1(X)) -> UNQUOTE11(X)
ACTIVE1(sel12(X1, X2)) -> SEL12(X1, active1(X2))
ACTIVE1(sel2(s1(X), cons2(Y, Z))) -> SEL2(X, Z)
PROPER1(s1(X)) -> S1(proper1(X))
CONS12(ok1(X1), ok1(X2)) -> CONS12(X1, X2)
ACTIVE1(fcons2(X1, X2)) -> FCONS2(X1, active1(X2))
S1(ok1(X)) -> S1(X)
ACTIVE1(first2(s1(X), cons2(Y, Z))) -> CONS2(Y, first2(X, Z))
ACTIVE1(from1(X)) -> CONS2(X, from1(s1(X)))
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons12(X1, X2)) -> CONS12(active1(X1), X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(first12(s1(X), cons2(Y, Z))) -> CONS12(quote1(Y), first12(X, Z))
PROPER1(cons12(X1, X2)) -> CONS12(proper1(X1), proper1(X2))
SEL12(ok1(X1), ok1(X2)) -> SEL12(X1, X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
ACTIVE1(from1(X)) -> S1(X)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
PROPER1(from1(X)) -> PROPER1(X)
ACTIVE1(sel2(X1, X2)) -> SEL2(active1(X1), X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
FCONS2(X1, mark1(X2)) -> FCONS2(X1, X2)
FROM1(ok1(X)) -> FROM1(X)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X1)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 17 SCCs with 58 less nodes.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

QUOTE11(ok1(X)) -> QUOTE11(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


QUOTE11(ok1(X)) -> QUOTE11(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( QUOTE11(x1) ) = max{0, x1 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

QUOTE1(ok1(X)) -> QUOTE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


QUOTE1(ok1(X)) -> QUOTE1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( QUOTE1(x1) ) = max{0, x1 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FCONS2(ok1(X1), ok1(X2)) -> FCONS2(X1, X2)
FCONS2(mark1(X1), X2) -> FCONS2(X1, X2)
FCONS2(X1, mark1(X2)) -> FCONS2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FCONS2(ok1(X1), ok1(X2)) -> FCONS2(X1, X2)
The remaining pairs can at least be oriented weakly.

FCONS2(mark1(X1), X2) -> FCONS2(X1, X2)
FCONS2(X1, mark1(X2)) -> FCONS2(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FCONS2(x1, x2) ) = max{0, x2 - 1}


POL( ok1(x1) ) = x1 + 2


POL( mark1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FCONS2(mark1(X1), X2) -> FCONS2(X1, X2)
FCONS2(X1, mark1(X2)) -> FCONS2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FCONS2(mark1(X1), X2) -> FCONS2(X1, X2)
The remaining pairs can at least be oriented weakly.

FCONS2(X1, mark1(X2)) -> FCONS2(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FCONS2(x1, x2) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FCONS2(X1, mark1(X2)) -> FCONS2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FCONS2(X1, mark1(X2)) -> FCONS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FCONS2(x1, x2) ) = max{0, x2 - 1}


POL( mark1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

UNQUOTE11(ok1(X)) -> UNQUOTE11(X)
UNQUOTE11(mark1(X)) -> UNQUOTE11(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


UNQUOTE11(ok1(X)) -> UNQUOTE11(X)
The remaining pairs can at least be oriented weakly.

UNQUOTE11(mark1(X)) -> UNQUOTE11(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( UNQUOTE11(x1) ) = max{0, x1 - 1}


POL( ok1(x1) ) = x1 + 2


POL( mark1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

UNQUOTE11(mark1(X)) -> UNQUOTE11(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


UNQUOTE11(mark1(X)) -> UNQUOTE11(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( UNQUOTE11(x1) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

UNQUOTE1(mark1(X)) -> UNQUOTE1(X)
UNQUOTE1(ok1(X)) -> UNQUOTE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


UNQUOTE1(mark1(X)) -> UNQUOTE1(X)
The remaining pairs can at least be oriented weakly.

UNQUOTE1(ok1(X)) -> UNQUOTE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( UNQUOTE1(x1) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

UNQUOTE1(ok1(X)) -> UNQUOTE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


UNQUOTE1(ok1(X)) -> UNQUOTE1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( UNQUOTE1(x1) ) = max{0, x1 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

S11(mark1(X)) -> S11(X)
S11(ok1(X)) -> S11(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


S11(mark1(X)) -> S11(X)
The remaining pairs can at least be oriented weakly.

S11(ok1(X)) -> S11(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( S11(x1) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

S11(ok1(X)) -> S11(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


S11(ok1(X)) -> S11(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( S11(x1) ) = max{0, x1 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

CONS12(ok1(X1), ok1(X2)) -> CONS12(X1, X2)
CONS12(mark1(X1), X2) -> CONS12(X1, X2)
CONS12(X1, mark1(X2)) -> CONS12(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONS12(ok1(X1), ok1(X2)) -> CONS12(X1, X2)
The remaining pairs can at least be oriented weakly.

CONS12(mark1(X1), X2) -> CONS12(X1, X2)
CONS12(X1, mark1(X2)) -> CONS12(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( CONS12(x1, x2) ) = max{0, x2 - 1}


POL( ok1(x1) ) = x1 + 2


POL( mark1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

CONS12(X1, mark1(X2)) -> CONS12(X1, X2)
CONS12(mark1(X1), X2) -> CONS12(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONS12(X1, mark1(X2)) -> CONS12(X1, X2)
The remaining pairs can at least be oriented weakly.

CONS12(mark1(X1), X2) -> CONS12(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( CONS12(x1, x2) ) = max{0, x2 - 1}


POL( mark1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

CONS12(mark1(X1), X2) -> CONS12(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONS12(mark1(X1), X2) -> CONS12(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( CONS12(x1, x2) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FIRST12(mark1(X1), X2) -> FIRST12(X1, X2)
FIRST12(X1, mark1(X2)) -> FIRST12(X1, X2)
FIRST12(ok1(X1), ok1(X2)) -> FIRST12(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FIRST12(mark1(X1), X2) -> FIRST12(X1, X2)
The remaining pairs can at least be oriented weakly.

FIRST12(X1, mark1(X2)) -> FIRST12(X1, X2)
FIRST12(ok1(X1), ok1(X2)) -> FIRST12(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FIRST12(x1, x2) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FIRST12(X1, mark1(X2)) -> FIRST12(X1, X2)
FIRST12(ok1(X1), ok1(X2)) -> FIRST12(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FIRST12(X1, mark1(X2)) -> FIRST12(X1, X2)
The remaining pairs can at least be oriented weakly.

FIRST12(ok1(X1), ok1(X2)) -> FIRST12(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FIRST12(x1, x2) ) = max{0, x2 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FIRST12(ok1(X1), ok1(X2)) -> FIRST12(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FIRST12(ok1(X1), ok1(X2)) -> FIRST12(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FIRST12(x1, x2) ) = max{0, x2 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

SEL12(mark1(X1), X2) -> SEL12(X1, X2)
SEL12(X1, mark1(X2)) -> SEL12(X1, X2)
SEL12(ok1(X1), ok1(X2)) -> SEL12(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SEL12(mark1(X1), X2) -> SEL12(X1, X2)
The remaining pairs can at least be oriented weakly.

SEL12(X1, mark1(X2)) -> SEL12(X1, X2)
SEL12(ok1(X1), ok1(X2)) -> SEL12(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( SEL12(x1, x2) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

SEL12(X1, mark1(X2)) -> SEL12(X1, X2)
SEL12(ok1(X1), ok1(X2)) -> SEL12(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SEL12(X1, mark1(X2)) -> SEL12(X1, X2)
The remaining pairs can at least be oriented weakly.

SEL12(ok1(X1), ok1(X2)) -> SEL12(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( SEL12(x1, x2) ) = max{0, x2 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

SEL12(ok1(X1), ok1(X2)) -> SEL12(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SEL12(ok1(X1), ok1(X2)) -> SEL12(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( SEL12(x1, x2) ) = max{0, x2 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FROM1(mark1(X)) -> FROM1(X)
FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FROM1(mark1(X)) -> FROM1(X)
The remaining pairs can at least be oriented weakly.

FROM1(ok1(X)) -> FROM1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FROM1(x1) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FROM1(ok1(X)) -> FROM1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FROM1(ok1(X)) -> FROM1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FROM1(x1) ) = max{0, x1 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FIRST2(X1, mark1(X2)) -> FIRST2(X1, X2)
FIRST2(ok1(X1), ok1(X2)) -> FIRST2(X1, X2)
FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FIRST2(X1, mark1(X2)) -> FIRST2(X1, X2)
The remaining pairs can at least be oriented weakly.

FIRST2(ok1(X1), ok1(X2)) -> FIRST2(X1, X2)
FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FIRST2(x1, x2) ) = max{0, x2 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FIRST2(ok1(X1), ok1(X2)) -> FIRST2(X1, X2)
FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FIRST2(ok1(X1), ok1(X2)) -> FIRST2(X1, X2)
The remaining pairs can at least be oriented weakly.

FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FIRST2(x1, x2) ) = max{0, x2 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


FIRST2(mark1(X1), X2) -> FIRST2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( FIRST2(x1, x2) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

CONS2(mark1(X1), X2) -> CONS2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONS2(mark1(X1), X2) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.

CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( CONS2(x1, x2) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( CONS2(x1, x2) ) = max{0, x2 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

S1(ok1(X)) -> S1(X)
S1(mark1(X)) -> S1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


S1(ok1(X)) -> S1(X)
The remaining pairs can at least be oriented weakly.

S1(mark1(X)) -> S1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( S1(x1) ) = max{0, x1 - 1}


POL( ok1(x1) ) = x1 + 2


POL( mark1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

S1(mark1(X)) -> S1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


S1(mark1(X)) -> S1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( S1(x1) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
The remaining pairs can at least be oriented weakly.

SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( SEL2(x1, x2) ) = max{0, x2 - 1}


POL( mark1(x1) ) = x1 + 2


POL( ok1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
The remaining pairs can at least be oriented weakly.

SEL2(mark1(X1), X2) -> SEL2(X1, X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( SEL2(x1, x2) ) = max{0, x2 - 1}


POL( ok1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

SEL2(mark1(X1), X2) -> SEL2(X1, X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


SEL2(mark1(X1), X2) -> SEL2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( SEL2(x1, x2) ) = max{0, x1 - 1}


POL( mark1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(quote1(X)) -> PROPER1(X)
PROPER1(unquote11(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons12(X1, X2)) -> PROPER1(X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(quote1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(unquote11(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons12(X1, X2)) -> PROPER1(X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( quote1(x1) ) = x1 + 2


POL( unquote11(x1) ) = x1


POL( sel2(x1, x2) ) = x1 + x2 + 1


POL( cons2(x1, x2) ) = x1 + x2 + 1


POL( cons12(x1, x2) ) = x1 + x2 + 1


POL( unquote1(x1) ) = x1


POL( s11(x1) ) = x1


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( sel12(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1


POL( s1(x1) ) = x1


POL( from1(x1) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(unquote11(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons12(X1, X2)) -> PROPER1(X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(unquote11(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons12(X1, X2)) -> PROPER1(X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( unquote11(x1) ) = x1 + 2


POL( sel2(x1, x2) ) = x1 + x2 + 1


POL( cons2(x1, x2) ) = x1 + x2 + 1


POL( cons12(x1, x2) ) = x1 + x2 + 1


POL( unquote1(x1) ) = x1


POL( s11(x1) ) = x1


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( sel12(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1


POL( s1(x1) ) = x1


POL( from1(x1) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons12(X1, X2)) -> PROPER1(X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons12(X1, X2)) -> PROPER1(X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( sel2(x1, x2) ) = x1 + x2 + 2


POL( cons2(x1, x2) ) = x1 + x2 + 1


POL( cons12(x1, x2) ) = x1 + x2 + 1


POL( unquote1(x1) ) = x1


POL( s11(x1) ) = x1


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( sel12(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1


POL( s1(x1) ) = x1


POL( from1(x1) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons12(X1, X2)) -> PROPER1(X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(cons12(X1, X2)) -> PROPER1(X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( cons2(x1, x2) ) = x1 + x2 + 2


POL( cons12(x1, x2) ) = x1 + x2 + 1


POL( unquote1(x1) ) = x1


POL( s11(x1) ) = x1


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( sel12(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1


POL( s1(x1) ) = x1


POL( from1(x1) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
QDP
                            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(cons12(X1, X2)) -> PROPER1(X2)
PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(cons12(X1, X2)) -> PROPER1(X2)
PROPER1(cons12(X1, X2)) -> PROPER1(X1)
The remaining pairs can at least be oriented weakly.

PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( cons12(x1, x2) ) = x1 + x2 + 2


POL( unquote1(x1) ) = x1


POL( s11(x1) ) = x1


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( sel12(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1


POL( s1(x1) ) = x1


POL( from1(x1) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
QDP
                                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(unquote1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(unquote1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( unquote1(x1) ) = x1 + 2


POL( s11(x1) ) = x1


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( sel12(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1


POL( s1(x1) ) = x1


POL( from1(x1) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
QDP
                                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(s1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(s1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(from1(X)) -> PROPER1(X)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( s1(x1) ) = x1 + 2


POL( from1(x1) ) = x1


POL( sel12(x1, x2) ) = x1 + x2 + 1


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( s11(x1) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
QDP
                                        ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(from1(X)) -> PROPER1(X)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(from1(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( from1(x1) ) = x1 + 2


POL( s11(x1) ) = x1


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( sel12(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
QDP
                                            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(sel12(X1, X2)) -> PROPER1(X1)
PROPER1(sel12(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(s11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( sel12(x1, x2) ) = x1 + x2 + 2


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( s11(x1) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
QDP
                                                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(s11(X)) -> PROPER1(X)
PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(s11(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.

PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( s11(x1) ) = x1 + 2


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
QDP
                                                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(first12(X1, X2)) -> PROPER1(X1)
PROPER1(first12(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( first12(x1, x2) ) = x1 + x2 + 2


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
QDP
                                                        ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(first2(X1, X2)) -> PROPER1(X1)
PROPER1(first2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( first2(x1, x2) ) = x1 + x2 + 2


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( quote11(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
QDP
                                                            ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(quote11(X)) -> PROPER1(X)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(fcons2(X1, X2)) -> PROPER1(X1)
PROPER1(fcons2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.

PROPER1(quote11(X)) -> PROPER1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( fcons2(x1, x2) ) = x1 + x2 + 2


POL( quote11(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
QDP
                                                                ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(quote11(X)) -> PROPER1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


PROPER1(quote11(X)) -> PROPER1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( PROPER1(x1) ) = max{0, x1 - 1}


POL( quote11(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
                                                          ↳ QDP
                                                            ↳ QDPOrderProof
                                                              ↳ QDP
                                                                ↳ QDPOrderProof
QDP
                                                                    ↳ PisEmptyProof
          ↳ QDP
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP
            ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X1)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least be oriented weakly.

ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X1)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( sel2(x1, x2) ) = x1 + x2 + 2


POL( sel12(x1, x2) ) = x1 + x2 + 1


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( cons2(x1, x2) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( unquote11(x1) ) = x1


POL( unquote1(x1) ) = x1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( cons12(x1, x2) ) = x1 + x2 + 1


POL( from1(x1) ) = x1


POL( s11(x1) ) = x1


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
QDP
                ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X2)
The remaining pairs can at least be oriented weakly.

ACTIVE1(first12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( sel12(x1, x2) ) = x1 + x2 + 2


POL( first12(x1, x2) ) = x1 + x2 + 1


POL( cons2(x1, x2) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( unquote11(x1) ) = x1


POL( unquote1(x1) ) = x1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( cons12(x1, x2) ) = x1 + x2 + 1


POL( from1(x1) ) = x1


POL( s11(x1) ) = x1


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(first12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X1)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(first12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(first12(X1, X2)) -> ACTIVE1(X2)
The remaining pairs can at least be oriented weakly.

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X1)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( first12(x1, x2) ) = x1 + x2 + 2


POL( cons2(x1, x2) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( unquote11(x1) ) = x1


POL( unquote1(x1) ) = x1


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( cons12(x1, x2) ) = x1 + x2 + 1


POL( from1(x1) ) = x1


POL( s11(x1) ) = x1


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
QDP
                        ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons12(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least be oriented weakly.

ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( cons12(x1, x2) ) = x1 + x2 + 2


POL( first2(x1, x2) ) = x1 + x2 + 1


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( cons2(x1, x2) ) = x1


POL( unquote1(x1) ) = x1


POL( unquote11(x1) ) = x1


POL( from1(x1) ) = x1


POL( s11(x1) ) = x1


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
QDP
                            ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(first2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(first2(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least be oriented weakly.

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( first2(x1, x2) ) = x1 + x2 + 2


POL( cons2(x1, x2) ) = x1


POL( fcons2(x1, x2) ) = x1 + x2 + 1


POL( unquote11(x1) ) = x1


POL( unquote1(x1) ) = x1


POL( from1(x1) ) = x1


POL( s11(x1) ) = x1


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
QDP
                                ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fcons2(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least be oriented weakly.

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( fcons2(x1, x2) ) = x1 + x2 + 2


POL( cons2(x1, x2) ) = x1


POL( unquote1(x1) ) = x1


POL( unquote11(x1) ) = x1


POL( from1(x1) ) = x1


POL( s11(x1) ) = x1


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
QDP
                                    ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least be oriented weakly.

ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( cons2(x1, x2) ) = x1 + 2


POL( unquote11(x1) ) = x1


POL( unquote1(x1) ) = x1


POL( from1(x1) ) = x1


POL( s11(x1) ) = x1


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
QDP
                                        ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(unquote1(X)) -> ACTIVE1(X)
ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(unquote1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( unquote1(x1) ) = x1 + 2


POL( unquote11(x1) ) = x1


POL( from1(x1) ) = x1


POL( s11(x1) ) = x1


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
QDP
                                            ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(unquote11(X)) -> ACTIVE1(X)
ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(unquote11(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( unquote11(x1) ) = x1 + 2


POL( from1(x1) ) = x1


POL( s11(x1) ) = x1


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
QDP
                                                ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(from1(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(from1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( from1(x1) ) = x1 + 2


POL( s11(x1) ) = x1


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
QDP
                                                    ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(s11(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(s11(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.

ACTIVE1(s1(X)) -> ACTIVE1(X)
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( s11(x1) ) = x1 + 2


POL( s1(x1) ) = x1



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
QDP
                                                        ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(s1(X)) -> ACTIVE1(X)

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].


The following pairs can be oriented strictly and are deleted.


ACTIVE1(s1(X)) -> ACTIVE1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial Order [17,21] with Interpretation:

POL( ACTIVE1(x1) ) = max{0, x1 - 1}


POL( s1(x1) ) = x1 + 2



The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
                  ↳ QDP
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
                          ↳ QDP
                            ↳ QDPOrderProof
                              ↳ QDP
                                ↳ QDPOrderProof
                                  ↳ QDP
                                    ↳ QDPOrderProof
                                      ↳ QDP
                                        ↳ QDPOrderProof
                                          ↳ QDP
                                            ↳ QDPOrderProof
                                              ↳ QDP
                                                ↳ QDPOrderProof
                                                  ↳ QDP
                                                    ↳ QDPOrderProof
                                                      ↳ QDP
                                                        ↳ QDPOrderProof
QDP
                                                            ↳ PisEmptyProof
          ↳ QDP

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

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
QDP

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

TOP1(ok1(X)) -> TOP1(active1(X))
TOP1(mark1(X)) -> TOP1(proper1(X))

The TRS R consists of the following rules:

active1(sel2(s1(X), cons2(Y, Z))) -> mark1(sel2(X, Z))
active1(sel2(0, cons2(X, Z))) -> mark1(X)
active1(first2(0, Z)) -> mark1(nil)
active1(first2(s1(X), cons2(Y, Z))) -> mark1(cons2(Y, first2(X, Z)))
active1(from1(X)) -> mark1(cons2(X, from1(s1(X))))
active1(sel12(s1(X), cons2(Y, Z))) -> mark1(sel12(X, Z))
active1(sel12(0, cons2(X, Z))) -> mark1(quote1(X))
active1(first12(0, Z)) -> mark1(nil1)
active1(first12(s1(X), cons2(Y, Z))) -> mark1(cons12(quote1(Y), first12(X, Z)))
active1(quote1(0)) -> mark1(01)
active1(quote11(cons2(X, Z))) -> mark1(cons12(quote1(X), quote11(Z)))
active1(quote11(nil)) -> mark1(nil1)
active1(quote1(s1(X))) -> mark1(s11(quote1(X)))
active1(quote1(sel2(X, Z))) -> mark1(sel12(X, Z))
active1(quote11(first2(X, Z))) -> mark1(first12(X, Z))
active1(unquote1(01)) -> mark1(0)
active1(unquote1(s11(X))) -> mark1(s1(unquote1(X)))
active1(unquote11(nil1)) -> mark1(nil)
active1(unquote11(cons12(X, Z))) -> mark1(fcons2(unquote1(X), unquote11(Z)))
active1(fcons2(X, Z)) -> mark1(cons2(X, Z))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(active1(X1), X2)
active1(first2(X1, X2)) -> first2(X1, active1(X2))
active1(from1(X)) -> from1(active1(X))
active1(sel12(X1, X2)) -> sel12(active1(X1), X2)
active1(sel12(X1, X2)) -> sel12(X1, active1(X2))
active1(first12(X1, X2)) -> first12(active1(X1), X2)
active1(first12(X1, X2)) -> first12(X1, active1(X2))
active1(cons12(X1, X2)) -> cons12(active1(X1), X2)
active1(cons12(X1, X2)) -> cons12(X1, active1(X2))
active1(s11(X)) -> s11(active1(X))
active1(unquote1(X)) -> unquote1(active1(X))
active1(unquote11(X)) -> unquote11(active1(X))
active1(fcons2(X1, X2)) -> fcons2(active1(X1), X2)
active1(fcons2(X1, X2)) -> fcons2(X1, active1(X2))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
first2(mark1(X1), X2) -> mark1(first2(X1, X2))
first2(X1, mark1(X2)) -> mark1(first2(X1, X2))
from1(mark1(X)) -> mark1(from1(X))
sel12(mark1(X1), X2) -> mark1(sel12(X1, X2))
sel12(X1, mark1(X2)) -> mark1(sel12(X1, X2))
first12(mark1(X1), X2) -> mark1(first12(X1, X2))
first12(X1, mark1(X2)) -> mark1(first12(X1, X2))
cons12(mark1(X1), X2) -> mark1(cons12(X1, X2))
cons12(X1, mark1(X2)) -> mark1(cons12(X1, X2))
s11(mark1(X)) -> mark1(s11(X))
unquote1(mark1(X)) -> mark1(unquote1(X))
unquote11(mark1(X)) -> mark1(unquote11(X))
fcons2(mark1(X1), X2) -> mark1(fcons2(X1, X2))
fcons2(X1, mark1(X2)) -> mark1(fcons2(X1, X2))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(0) -> ok1(0)
proper1(first2(X1, X2)) -> first2(proper1(X1), proper1(X2))
proper1(nil) -> ok1(nil)
proper1(from1(X)) -> from1(proper1(X))
proper1(sel12(X1, X2)) -> sel12(proper1(X1), proper1(X2))
proper1(quote1(X)) -> quote1(proper1(X))
proper1(first12(X1, X2)) -> first12(proper1(X1), proper1(X2))
proper1(nil1) -> ok1(nil1)
proper1(cons12(X1, X2)) -> cons12(proper1(X1), proper1(X2))
proper1(01) -> ok1(01)
proper1(quote11(X)) -> quote11(proper1(X))
proper1(s11(X)) -> s11(proper1(X))
proper1(unquote1(X)) -> unquote1(proper1(X))
proper1(unquote11(X)) -> unquote11(proper1(X))
proper1(fcons2(X1, X2)) -> fcons2(proper1(X1), proper1(X2))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
first2(ok1(X1), ok1(X2)) -> ok1(first2(X1, X2))
from1(ok1(X)) -> ok1(from1(X))
sel12(ok1(X1), ok1(X2)) -> ok1(sel12(X1, X2))
quote1(ok1(X)) -> ok1(quote1(X))
first12(ok1(X1), ok1(X2)) -> ok1(first12(X1, X2))
cons12(ok1(X1), ok1(X2)) -> ok1(cons12(X1, X2))
quote11(ok1(X)) -> ok1(quote11(X))
s11(ok1(X)) -> ok1(s11(X))
unquote1(ok1(X)) -> ok1(unquote1(X))
unquote11(ok1(X)) -> ok1(unquote11(X))
fcons2(ok1(X1), ok1(X2)) -> ok1(fcons2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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