Termination w.r.t. Q proof of TRS/TRCSREx6_15_AEL02

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

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

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) ) = x₁

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( ok₁(x₁) ) = x₁ + 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

            ↳ 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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 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

            ↳ 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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( ok₁(x₁) ) = x₁ + 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

            ↳ 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) ) = x₁

POL( ok₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 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

            ↳ 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) ) = x₂

POL( mark₁(x₁) ) = x₁ + 1

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) ) = x₂

POL( ok₁(x₁) ) = x₁ + 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

            ↳ 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) ) = x₁

POL( mark₁(x₁) ) = x₁ + 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

            ↳ 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) ) = x₁

POL( quote₁(x₁) ) = x₁ + 1

POL( unquote1₁(x₁) ) = x₁

POL( sel₂(x₁, x₂) ) = x₁ + x₂

POL( cons₂(x₁, x₂) ) = x₁ + x₂

POL( cons1₂(x₁, x₂) ) = x₁ + x₂

POL( unquote₁(x₁) ) = x₁

POL( s1₁(x₁) ) = x₁

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( sel1₂(x₁, x₂) ) = x₁ + x₂

POL( quote1₁(x₁) ) = x₁

POL( s₁(x₁) ) = x₁

POL( from₁(x₁) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( unquote1₁(x₁) ) = x₁ + 1

POL( sel₂(x₁, x₂) ) = x₁ + x₂

POL( cons₂(x₁, x₂) ) = x₁ + x₂

POL( cons1₂(x₁, x₂) ) = x₁ + x₂

POL( unquote₁(x₁) ) = x₁

POL( s1₁(x₁) ) = x₁

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( sel1₂(x₁, x₂) ) = x₁ + x₂

POL( quote1₁(x₁) ) = x₁

POL( s₁(x₁) ) = x₁

POL( from₁(x₁) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( sel₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( cons₂(x₁, x₂) ) = x₁ + x₂

POL( cons1₂(x₁, x₂) ) = x₁ + x₂

POL( unquote₁(x₁) ) = x₁

POL( s1₁(x₁) ) = x₁

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( sel1₂(x₁, x₂) ) = x₁ + x₂

POL( quote1₁(x₁) ) = x₁

POL( s₁(x₁) ) = x₁

POL( from₁(x₁) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( cons₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( cons1₂(x₁, x₂) ) = x₁ + x₂

POL( unquote₁(x₁) ) = x₁

POL( s1₁(x₁) ) = x₁

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( sel1₂(x₁, x₂) ) = x₁ + x₂

POL( quote1₁(x₁) ) = x₁

POL( s₁(x₁) ) = x₁

POL( from₁(x₁) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( cons1₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( unquote₁(x₁) ) = x₁

POL( s1₁(x₁) ) = x₁

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( sel1₂(x₁, x₂) ) = x₁ + x₂

POL( quote1₁(x₁) ) = x₁

POL( s₁(x₁) ) = x₁

POL( from₁(x₁) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( unquote₁(x₁) ) = x₁ + 1

POL( s1₁(x₁) ) = x₁

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( sel1₂(x₁, x₂) ) = x₁ + x₂

POL( quote1₁(x₁) ) = x₁

POL( s₁(x₁) ) = x₁

POL( from₁(x₁) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( s₁(x₁) ) = x₁ + 1

POL( from₁(x₁) ) = x₁

POL( sel1₂(x₁, x₂) ) = x₁ + x₂

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( s1₁(x₁) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( from₁(x₁) ) = x₁ + 1

POL( s1₁(x₁) ) = x₁

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( sel1₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( sel1₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( s1₁(x₁) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( s1₁(x₁) ) = x₁ + 1

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( first1₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

POL( first₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( first₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

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

                                                                ↳ 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) ) = x₁

POL( quote1₁(x₁) ) = x₁ + 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

                                        ↳ 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) ) = x₁

POL( sel₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( sel1₂(x₁, x₂) ) = x₁ + x₂

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( cons₂(x₁, x₂) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

POL( unquote1₁(x₁) ) = x₁

POL( unquote₁(x₁) ) = x₁

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( cons1₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( sel1₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( first1₂(x₁, x₂) ) = x₁ + x₂

POL( cons₂(x₁, x₂) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

POL( unquote1₁(x₁) ) = x₁

POL( unquote₁(x₁) ) = x₁

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( cons1₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( first1₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( cons₂(x₁, x₂) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

POL( unquote1₁(x₁) ) = x₁

POL( unquote₁(x₁) ) = x₁

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( cons1₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( cons1₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( first₂(x₁, x₂) ) = x₁ + x₂

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

POL( first₂(x₁, x₂) ) = x₁ + x₂ + 1

POL( cons₂(x₁, x₂) ) = x₁

POL( fcons₂(x₁, x₂) ) = x₁ + x₂

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) ) = x₁

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

                                ↳ 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) ) = x₁

POL( cons₂(x₁, x₂) ) = x₁ + 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

                                    ↳ 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) ) = x₁

POL( unquote₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( unquote1₁(x₁) ) = x₁ + 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

                            ↳ 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) ) = x₁

POL( from₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( s1₁(x₁) ) = x₁ + 1

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) ) = x₁

POL( s₁(x₁) ) = x₁ + 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

          ↳ 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.