Termination w.r.t. Q proof of TRS/TRCSREx5_7_Luc97

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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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:

PROPER₁(from₁(X)) -> FROM₁(proper₁(X))
ACTIVE₁(from₁(X)) -> FROM₁(s₁(X))
ACTIVE₁(sel₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(quote₁(X)) -> ACTIVE₁(X)
ACTIVE₁(quote₁(sel₂(X, Y))) -> SEL1₂(X, Y)
ACTIVE₁(sel1₂(s₁(X), cons₂(Y, Z))) -> SEL1₂(X, Z)
SEL1₂(mark₁(X1), X2) -> SEL1₂(X1, X2)
ACTIVE₁(sel1₂(X1, X2)) -> ACTIVE₁(X2)
PROPER₁(s1₁(X)) -> PROPER₁(X)
ACTIVE₁(dbl₁(s₁(X))) -> DBL₁(X)
PROPER₁(quote₁(X)) -> QUOTE₁(proper₁(X))
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))
ACTIVE₁(quote₁(s₁(X))) -> QUOTE₁(X)
ACTIVE₁(sel1₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(sel1₂(X1, X2)) -> SEL1₂(proper₁(X1), proper₁(X2))
PROPER₁(s1₁(X)) -> S1₁(proper₁(X))
PROPER₁(indx₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(dbls₁(cons₂(X, Y))) -> DBLS₁(Y)
PROPER₁(sel1₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(dbls₁(X)) -> DBLS₁(proper₁(X))
PROPER₁(sel₂(X1, X2)) -> PROPER₁(X2)
TOP₁(ok₁(X)) -> ACTIVE₁(X)
ACTIVE₁(quote₁(s₁(X))) -> S1₁(quote₁(X))
SEL₂(mark₁(X1), X2) -> SEL₂(X1, X2)
ACTIVE₁(indx₂(X1, X2)) -> INDX₂(active₁(X1), X2)
DBL1₁(ok₁(X)) -> DBL1₁(X)
ACTIVE₁(indx₂(cons₂(X, Y), Z)) -> INDX₂(Y, Z)
PROPER₁(s₁(X)) -> PROPER₁(X)
ACTIVE₁(dbl1₁(s₁(X))) -> DBL1₁(X)
ACTIVE₁(dbls₁(X)) -> DBLS₁(active₁(X))
ACTIVE₁(quote₁(X)) -> QUOTE₁(active₁(X))
ACTIVE₁(dbls₁(X)) -> ACTIVE₁(X)
PROPER₁(indx₂(X1, X2)) -> PROPER₁(X1)
ACTIVE₁(sel1₂(X1, X2)) -> SEL1₂(active₁(X1), X2)
PROPER₁(dbl1₁(X)) -> PROPER₁(X)
ACTIVE₁(dbls₁(cons₂(X, Y))) -> DBL₁(X)
ACTIVE₁(dbl1₁(X)) -> ACTIVE₁(X)
TOP₁(mark₁(X)) -> PROPER₁(X)
ACTIVE₁(dbl1₁(X)) -> DBL1₁(active₁(X))
SEL1₂(X1, mark₁(X2)) -> SEL1₂(X1, X2)
DBL₁(ok₁(X)) -> DBL₁(X)
PROPER₁(dbl1₁(X)) -> DBL1₁(proper₁(X))
PROPER₁(sel₂(X1, X2)) -> SEL₂(proper₁(X1), proper₁(X2))
S1₁(ok₁(X)) -> S1₁(X)
DBLS₁(mark₁(X)) -> DBLS₁(X)
ACTIVE₁(quote₁(dbl₁(X))) -> DBL1₁(X)
ACTIVE₁(dbl₁(X)) -> DBL₁(active₁(X))
ACTIVE₁(sel₂(X1, X2)) -> SEL₂(X1, active₁(X2))
ACTIVE₁(sel1₂(X1, X2)) -> SEL1₂(X1, active₁(X2))
ACTIVE₁(sel₂(s₁(X), cons₂(Y, Z))) -> SEL₂(X, Z)
ACTIVE₁(sel₂(X1, X2)) -> ACTIVE₁(X1)
SEL₂(X1, mark₁(X2)) -> SEL₂(X1, X2)
PROPER₁(s₁(X)) -> S₁(proper₁(X))
PROPER₁(indx₂(X1, X2)) -> INDX₂(proper₁(X1), proper₁(X2))
PROPER₁(dbls₁(X)) -> PROPER₁(X)
ACTIVE₁(indx₂(cons₂(X, Y), Z)) -> CONS₂(sel₂(X, Z), indx₂(Y, Z))
ACTIVE₁(dbl1₁(s₁(X))) -> S1₁(s1₁(dbl1₁(X)))
ACTIVE₁(s1₁(X)) -> ACTIVE₁(X)
S₁(ok₁(X)) -> S₁(X)
QUOTE₁(ok₁(X)) -> QUOTE₁(X)
S1₁(mark₁(X)) -> S1₁(X)
INDX₂(mark₁(X1), X2) -> INDX₂(X1, X2)
ACTIVE₁(from₁(X)) -> CONS₂(X, from₁(s₁(X)))
ACTIVE₁(indx₂(X1, X2)) -> ACTIVE₁(X1)
PROPER₁(quote₁(X)) -> PROPER₁(X)
DBLS₁(ok₁(X)) -> DBLS₁(X)
PROPER₁(sel₂(X1, X2)) -> PROPER₁(X1)
DBL₁(mark₁(X)) -> DBL₁(X)
SEL₂(ok₁(X1), ok₁(X2)) -> SEL₂(X1, X2)
ACTIVE₁(dbl1₁(s₁(X))) -> S1₁(dbl1₁(X))
SEL1₂(ok₁(X1), ok₁(X2)) -> SEL1₂(X1, X2)
ACTIVE₁(dbl₁(s₁(X))) -> S₁(dbl₁(X))
ACTIVE₁(dbls₁(cons₂(X, Y))) -> CONS₂(dbl₁(X), dbls₁(Y))
ACTIVE₁(s1₁(X)) -> S1₁(active₁(X))
ACTIVE₁(from₁(X)) -> S₁(X)
DBL1₁(mark₁(X)) -> DBL1₁(X)
QUOTE₁(mark₁(X)) -> QUOTE₁(X)
PROPER₁(cons₂(X1, X2)) -> CONS₂(proper₁(X1), proper₁(X2))
PROPER₁(dbl₁(X)) -> DBL₁(proper₁(X))
TOP₁(ok₁(X)) -> TOP₁(active₁(X))
PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(sel₂(X1, X2)) -> SEL₂(active₁(X1), X2)
PROPER₁(sel1₂(X1, X2)) -> PROPER₁(X1)
CONS₂(ok₁(X1), ok₁(X2)) -> CONS₂(X1, X2)
FROM₁(ok₁(X)) -> FROM₁(X)
INDX₂(ok₁(X1), ok₁(X2)) -> INDX₂(X1, X2)
ACTIVE₁(indx₂(cons₂(X, Y), Z)) -> SEL₂(X, Z)
ACTIVE₁(dbl₁(s₁(X))) -> S₁(s₁(dbl₁(X)))
PROPER₁(dbl₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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:

PROPER₁(from₁(X)) -> FROM₁(proper₁(X))
ACTIVE₁(from₁(X)) -> FROM₁(s₁(X))
ACTIVE₁(sel₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(quote₁(X)) -> ACTIVE₁(X)
ACTIVE₁(quote₁(sel₂(X, Y))) -> SEL1₂(X, Y)
ACTIVE₁(sel1₂(s₁(X), cons₂(Y, Z))) -> SEL1₂(X, Z)
SEL1₂(mark₁(X1), X2) -> SEL1₂(X1, X2)
ACTIVE₁(sel1₂(X1, X2)) -> ACTIVE₁(X2)
PROPER₁(s1₁(X)) -> PROPER₁(X)
ACTIVE₁(dbl₁(s₁(X))) -> DBL₁(X)
PROPER₁(quote₁(X)) -> QUOTE₁(proper₁(X))
TOP₁(mark₁(X)) -> TOP₁(proper₁(X))
ACTIVE₁(quote₁(s₁(X))) -> QUOTE₁(X)
ACTIVE₁(sel1₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(sel1₂(X1, X2)) -> SEL1₂(proper₁(X1), proper₁(X2))
PROPER₁(s1₁(X)) -> S1₁(proper₁(X))
PROPER₁(indx₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(dbls₁(cons₂(X, Y))) -> DBLS₁(Y)
PROPER₁(sel1₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(dbls₁(X)) -> DBLS₁(proper₁(X))
PROPER₁(sel₂(X1, X2)) -> PROPER₁(X2)
TOP₁(ok₁(X)) -> ACTIVE₁(X)
ACTIVE₁(quote₁(s₁(X))) -> S1₁(quote₁(X))
SEL₂(mark₁(X1), X2) -> SEL₂(X1, X2)
ACTIVE₁(indx₂(X1, X2)) -> INDX₂(active₁(X1), X2)
DBL1₁(ok₁(X)) -> DBL1₁(X)
ACTIVE₁(indx₂(cons₂(X, Y), Z)) -> INDX₂(Y, Z)
PROPER₁(s₁(X)) -> PROPER₁(X)
ACTIVE₁(dbl1₁(s₁(X))) -> DBL1₁(X)
ACTIVE₁(dbls₁(X)) -> DBLS₁(active₁(X))
ACTIVE₁(quote₁(X)) -> QUOTE₁(active₁(X))
ACTIVE₁(dbls₁(X)) -> ACTIVE₁(X)
PROPER₁(indx₂(X1, X2)) -> PROPER₁(X1)
ACTIVE₁(sel1₂(X1, X2)) -> SEL1₂(active₁(X1), X2)
PROPER₁(dbl1₁(X)) -> PROPER₁(X)
ACTIVE₁(dbls₁(cons₂(X, Y))) -> DBL₁(X)
ACTIVE₁(dbl1₁(X)) -> ACTIVE₁(X)
TOP₁(mark₁(X)) -> PROPER₁(X)
ACTIVE₁(dbl1₁(X)) -> DBL1₁(active₁(X))
SEL1₂(X1, mark₁(X2)) -> SEL1₂(X1, X2)
DBL₁(ok₁(X)) -> DBL₁(X)
PROPER₁(dbl1₁(X)) -> DBL1₁(proper₁(X))
PROPER₁(sel₂(X1, X2)) -> SEL₂(proper₁(X1), proper₁(X2))
S1₁(ok₁(X)) -> S1₁(X)
DBLS₁(mark₁(X)) -> DBLS₁(X)
ACTIVE₁(quote₁(dbl₁(X))) -> DBL1₁(X)
ACTIVE₁(dbl₁(X)) -> DBL₁(active₁(X))
ACTIVE₁(sel₂(X1, X2)) -> SEL₂(X1, active₁(X2))
ACTIVE₁(sel1₂(X1, X2)) -> SEL1₂(X1, active₁(X2))
ACTIVE₁(sel₂(s₁(X), cons₂(Y, Z))) -> SEL₂(X, Z)
ACTIVE₁(sel₂(X1, X2)) -> ACTIVE₁(X1)
SEL₂(X1, mark₁(X2)) -> SEL₂(X1, X2)
PROPER₁(s₁(X)) -> S₁(proper₁(X))
PROPER₁(indx₂(X1, X2)) -> INDX₂(proper₁(X1), proper₁(X2))
PROPER₁(dbls₁(X)) -> PROPER₁(X)
ACTIVE₁(indx₂(cons₂(X, Y), Z)) -> CONS₂(sel₂(X, Z), indx₂(Y, Z))
ACTIVE₁(dbl1₁(s₁(X))) -> S1₁(s1₁(dbl1₁(X)))
ACTIVE₁(s1₁(X)) -> ACTIVE₁(X)
S₁(ok₁(X)) -> S₁(X)
QUOTE₁(ok₁(X)) -> QUOTE₁(X)
S1₁(mark₁(X)) -> S1₁(X)
INDX₂(mark₁(X1), X2) -> INDX₂(X1, X2)
ACTIVE₁(from₁(X)) -> CONS₂(X, from₁(s₁(X)))
ACTIVE₁(indx₂(X1, X2)) -> ACTIVE₁(X1)
PROPER₁(quote₁(X)) -> PROPER₁(X)
DBLS₁(ok₁(X)) -> DBLS₁(X)
PROPER₁(sel₂(X1, X2)) -> PROPER₁(X1)
DBL₁(mark₁(X)) -> DBL₁(X)
SEL₂(ok₁(X1), ok₁(X2)) -> SEL₂(X1, X2)
ACTIVE₁(dbl1₁(s₁(X))) -> S1₁(dbl1₁(X))
SEL1₂(ok₁(X1), ok₁(X2)) -> SEL1₂(X1, X2)
ACTIVE₁(dbl₁(s₁(X))) -> S₁(dbl₁(X))
ACTIVE₁(dbls₁(cons₂(X, Y))) -> CONS₂(dbl₁(X), dbls₁(Y))
ACTIVE₁(s1₁(X)) -> S1₁(active₁(X))
ACTIVE₁(from₁(X)) -> S₁(X)
DBL1₁(mark₁(X)) -> DBL1₁(X)
QUOTE₁(mark₁(X)) -> QUOTE₁(X)
PROPER₁(cons₂(X1, X2)) -> CONS₂(proper₁(X1), proper₁(X2))
PROPER₁(dbl₁(X)) -> DBL₁(proper₁(X))
TOP₁(ok₁(X)) -> TOP₁(active₁(X))
PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
ACTIVE₁(sel₂(X1, X2)) -> SEL₂(active₁(X1), X2)
PROPER₁(sel1₂(X1, X2)) -> PROPER₁(X1)
CONS₂(ok₁(X1), ok₁(X2)) -> CONS₂(X1, X2)
FROM₁(ok₁(X)) -> FROM₁(X)
INDX₂(ok₁(X1), ok₁(X2)) -> INDX₂(X1, X2)
ACTIVE₁(indx₂(cons₂(X, Y), Z)) -> SEL₂(X, Z)
ACTIVE₁(dbl₁(s₁(X))) -> S₁(s₁(dbl₁(X)))
PROPER₁(dbl₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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 14 SCCs with 44 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ 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:

FROM₁(ok₁(X)) -> FROM₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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 interpretation [21]:

POL(FROM₁(x₁)) = x₁
POL(ok₁(x₁)) = 1 + x₁

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

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

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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 interpretation [21]:

POL(CONS₂(x₁, x₂)) = x₂
POL(ok₁(x₁)) = 1 + x₁

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

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

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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

S₁(ok₁(X)) -> S₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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.
none
Used ordering: Polynomial interpretation [21]:

POL(S₁(x₁)) = x₁
POL(ok₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ 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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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

QUOTE₁(mark₁(X)) -> QUOTE₁(X)
QUOTE₁(ok₁(X)) -> QUOTE₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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(mark₁(X)) -> QUOTE₁(X)
QUOTE₁(ok₁(X)) -> QUOTE₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(QUOTE₁(x₁)) = x₁
POL(mark₁(x₁)) = 1 + x₁
POL(ok₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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)
SEL1₂(ok₁(X1), ok₁(X2)) -> SEL1₂(X1, X2)

The remaining pairs can at least be oriented weakly.

SEL1₂(mark₁(X1), X2) -> SEL1₂(X1, X2)

Used ordering: Polynomial interpretation [21]:

POL(SEL1₂(x₁, x₂)) = x₂
POL(mark₁(x₁)) = 1 + x₁
POL(ok₁(x₁)) = 1 + 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

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

SEL1₂(mark₁(X1), X2) -> SEL1₂(X1, X2)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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.
none
Used ordering: Polynomial interpretation [21]:

POL(SEL1₂(x₁, x₂)) = x₁
POL(mark₁(x₁)) = 1 + x₁

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

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

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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)
S1₁(ok₁(X)) -> S1₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(S1₁(x₁)) = x₁
POL(mark₁(x₁)) = 1 + x₁
POL(ok₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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

DBL1₁(ok₁(X)) -> DBL1₁(X)
DBL1₁(mark₁(X)) -> DBL1₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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.

DBL1₁(ok₁(X)) -> DBL1₁(X)
DBL1₁(mark₁(X)) -> DBL1₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(DBL1₁(x₁)) = x₁
POL(mark₁(x₁)) = 1 + x₁
POL(ok₁(x₁)) = 1 + x₁

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ 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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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

INDX₂(ok₁(X1), ok₁(X2)) -> INDX₂(X1, X2)
INDX₂(mark₁(X1), X2) -> INDX₂(X1, X2)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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.

INDX₂(ok₁(X1), ok₁(X2)) -> INDX₂(X1, X2)

The remaining pairs can at least be oriented weakly.

INDX₂(mark₁(X1), X2) -> INDX₂(X1, X2)

Used ordering: Polynomial interpretation [21]:

POL(INDX₂(x₁, x₂)) = x₂
POL(mark₁(x₁)) = 0
POL(ok₁(x₁)) = 1 + 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

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

INDX₂(mark₁(X1), X2) -> INDX₂(X1, X2)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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.

INDX₂(mark₁(X1), X2) -> INDX₂(X1, X2)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(INDX₂(x₁, x₂)) = x₁
POL(mark₁(x₁)) = 1 + 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

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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)
SEL₂(mark₁(X1), X2) -> SEL₂(X1, X2)

The remaining pairs can at least be oriented weakly.

SEL₂(X1, mark₁(X2)) -> SEL₂(X1, X2)

Used ordering: Polynomial interpretation [21]:

POL(SEL₂(x₁, x₂)) = x₁
POL(mark₁(x₁)) = 1 + x₁
POL(ok₁(x₁)) = 1 + 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

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

SEL₂(X1, mark₁(X2)) -> SEL₂(X1, X2)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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.
none
Used ordering: Polynomial interpretation [21]:

POL(SEL₂(x₁, x₂)) = x₂
POL(mark₁(x₁)) = 1 + 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

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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

DBLS₁(ok₁(X)) -> DBLS₁(X)
DBLS₁(mark₁(X)) -> DBLS₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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.

DBLS₁(ok₁(X)) -> DBLS₁(X)
DBLS₁(mark₁(X)) -> DBLS₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(DBLS₁(x₁)) = x₁
POL(mark₁(x₁)) = 1 + x₁
POL(ok₁(x₁)) = 1 + 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

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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

DBL₁(mark₁(X)) -> DBL₁(X)
DBL₁(ok₁(X)) -> DBL₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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.

DBL₁(mark₁(X)) -> DBL₁(X)
DBL₁(ok₁(X)) -> DBL₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(DBL₁(x₁)) = x₁
POL(mark₁(x₁)) = 1 + x₁
POL(ok₁(x₁)) = 1 + 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

                ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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

PROPER₁(quote₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(sel₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(s1₁(X)) -> PROPER₁(X)
PROPER₁(indx₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(sel1₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(sel₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(sel1₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(dbls₁(X)) -> PROPER₁(X)
PROPER₁(indx₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(dbl1₁(X)) -> PROPER₁(X)
PROPER₁(dbl₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(sel₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(s1₁(X)) -> PROPER₁(X)
PROPER₁(indx₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(sel₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(s₁(X)) -> PROPER₁(X)
PROPER₁(from₁(X)) -> PROPER₁(X)
PROPER₁(cons₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(dbls₁(X)) -> PROPER₁(X)
PROPER₁(indx₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(dbl₁(X)) -> PROPER₁(X)

The remaining pairs can at least be oriented weakly.

PROPER₁(sel1₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(sel1₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(dbl1₁(X)) -> PROPER₁(X)

Used ordering: Polynomial interpretation [21]:

POL(PROPER₁(x₁)) = x₁
POL(cons₂(x₁, x₂)) = 1 + x₁ + x₂
POL(dbl₁(x₁)) = 1 + x₁
POL(dbl1₁(x₁)) = x₁
POL(dbls₁(x₁)) = 1 + x₁
POL(from₁(x₁)) = 1 + x₁
POL(indx₂(x₁, x₂)) = 1 + x₁ + x₂
POL(quote₁(x₁)) = 1 + x₁
POL(s₁(x₁)) = 1 + x₁
POL(s1₁(x₁)) = 1 + x₁
POL(sel₂(x₁, x₂)) = 1 + x₁ + x₂
POL(sel1₂(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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

          ↳ QDP

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

PROPER₁(sel1₂(X1, X2)) -> PROPER₁(X1)
PROPER₁(sel1₂(X1, X2)) -> PROPER₁(X2)
PROPER₁(dbl1₁(X)) -> PROPER₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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)
PROPER₁(dbl1₁(X)) -> PROPER₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PROPER₁(x₁)) = x₁
POL(dbl1₁(x₁)) = 1 + x₁
POL(sel1₂(x₁, x₂)) = 1 + 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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ PisEmptyProof

          ↳ QDP

          ↳ QDP

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

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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

ACTIVE₁(sel1₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(dbl1₁(X)) -> ACTIVE₁(X)
ACTIVE₁(sel₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(quote₁(X)) -> ACTIVE₁(X)
ACTIVE₁(indx₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
ACTIVE₁(sel1₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(sel₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(dbls₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s1₁(X)) -> ACTIVE₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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(dbl1₁(X)) -> ACTIVE₁(X)
ACTIVE₁(sel₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(quote₁(X)) -> ACTIVE₁(X)
ACTIVE₁(indx₂(X1, X2)) -> ACTIVE₁(X1)
ACTIVE₁(sel1₂(X1, X2)) -> ACTIVE₁(X2)
ACTIVE₁(sel₂(X1, X2)) -> ACTIVE₁(X1)

The remaining pairs can at least be oriented weakly.

ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
ACTIVE₁(dbls₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s1₁(X)) -> ACTIVE₁(X)

Used ordering: Polynomial interpretation [21]:

POL(ACTIVE₁(x₁)) = x₁
POL(dbl₁(x₁)) = x₁
POL(dbl1₁(x₁)) = 1 + x₁
POL(dbls₁(x₁)) = x₁
POL(indx₂(x₁, x₂)) = 1 + x₁
POL(quote₁(x₁)) = 1 + x₁
POL(s1₁(x₁)) = x₁
POL(sel₂(x₁, x₂)) = 1 + x₁ + x₂
POL(sel1₂(x₁, x₂)) = 1 + 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

            ↳ QDPOrderProof

              ↳ QDP

                ↳ QDPOrderProof

          ↳ QDP

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

ACTIVE₁(dbl₁(X)) -> ACTIVE₁(X)
ACTIVE₁(dbls₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s1₁(X)) -> ACTIVE₁(X)

The TRS R consists of the following rules:

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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(dbl₁(X)) -> ACTIVE₁(X)
ACTIVE₁(dbls₁(X)) -> ACTIVE₁(X)
ACTIVE₁(s1₁(X)) -> ACTIVE₁(X)

The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE₁(x₁)) = x₁
POL(dbl₁(x₁)) = 1 + x₁
POL(dbls₁(x₁)) = 1 + x₁
POL(s1₁(x₁)) = 1 + 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

                    ↳ PisEmptyProof

          ↳ QDP

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

active₁(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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

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(dbl₁(0)) -> mark₁(0)
active₁(dbl₁(s₁(X))) -> mark₁(s₁(s₁(dbl₁(X))))
active₁(dbls₁(nil)) -> mark₁(nil)
active₁(dbls₁(cons₂(X, Y))) -> mark₁(cons₂(dbl₁(X), dbls₁(Y)))
active₁(sel₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel₂(X, Z))
active₁(indx₂(nil, X)) -> mark₁(nil)
active₁(indx₂(cons₂(X, Y), Z)) -> mark₁(cons₂(sel₂(X, Z), indx₂(Y, Z)))
active₁(from₁(X)) -> mark₁(cons₂(X, from₁(s₁(X))))
active₁(dbl1₁(0)) -> mark₁(01)
active₁(dbl1₁(s₁(X))) -> mark₁(s1₁(s1₁(dbl1₁(X))))
active₁(sel1₂(0, cons₂(X, Y))) -> mark₁(X)
active₁(sel1₂(s₁(X), cons₂(Y, Z))) -> mark₁(sel1₂(X, Z))
active₁(quote₁(0)) -> mark₁(01)
active₁(quote₁(s₁(X))) -> mark₁(s1₁(quote₁(X)))
active₁(quote₁(dbl₁(X))) -> mark₁(dbl1₁(X))
active₁(quote₁(sel₂(X, Y))) -> mark₁(sel1₂(X, Y))
active₁(dbl₁(X)) -> dbl₁(active₁(X))
active₁(dbls₁(X)) -> dbls₁(active₁(X))
active₁(sel₂(X1, X2)) -> sel₂(active₁(X1), X2)
active₁(sel₂(X1, X2)) -> sel₂(X1, active₁(X2))
active₁(indx₂(X1, X2)) -> indx₂(active₁(X1), X2)
active₁(dbl1₁(X)) -> dbl1₁(active₁(X))
active₁(s1₁(X)) -> s1₁(active₁(X))
active₁(sel1₂(X1, X2)) -> sel1₂(active₁(X1), X2)
active₁(sel1₂(X1, X2)) -> sel1₂(X1, active₁(X2))
active₁(quote₁(X)) -> quote₁(active₁(X))
dbl₁(mark₁(X)) -> mark₁(dbl₁(X))
dbls₁(mark₁(X)) -> mark₁(dbls₁(X))
sel₂(mark₁(X1), X2) -> mark₁(sel₂(X1, X2))
sel₂(X1, mark₁(X2)) -> mark₁(sel₂(X1, X2))
indx₂(mark₁(X1), X2) -> mark₁(indx₂(X1, X2))
dbl1₁(mark₁(X)) -> mark₁(dbl1₁(X))
s1₁(mark₁(X)) -> mark₁(s1₁(X))
sel1₂(mark₁(X1), X2) -> mark₁(sel1₂(X1, X2))
sel1₂(X1, mark₁(X2)) -> mark₁(sel1₂(X1, X2))
quote₁(mark₁(X)) -> mark₁(quote₁(X))
proper₁(dbl₁(X)) -> dbl₁(proper₁(X))
proper₁(0) -> ok₁(0)
proper₁(s₁(X)) -> s₁(proper₁(X))
proper₁(dbls₁(X)) -> dbls₁(proper₁(X))
proper₁(nil) -> ok₁(nil)
proper₁(cons₂(X1, X2)) -> cons₂(proper₁(X1), proper₁(X2))
proper₁(sel₂(X1, X2)) -> sel₂(proper₁(X1), proper₁(X2))
proper₁(indx₂(X1, X2)) -> indx₂(proper₁(X1), proper₁(X2))
proper₁(from₁(X)) -> from₁(proper₁(X))
proper₁(dbl1₁(X)) -> dbl1₁(proper₁(X))
proper₁(01) -> ok₁(01)
proper₁(s1₁(X)) -> s1₁(proper₁(X))
proper₁(sel1₂(X1, X2)) -> sel1₂(proper₁(X1), proper₁(X2))
proper₁(quote₁(X)) -> quote₁(proper₁(X))
dbl₁(ok₁(X)) -> ok₁(dbl₁(X))
s₁(ok₁(X)) -> ok₁(s₁(X))
dbls₁(ok₁(X)) -> ok₁(dbls₁(X))
cons₂(ok₁(X1), ok₁(X2)) -> ok₁(cons₂(X1, X2))
sel₂(ok₁(X1), ok₁(X2)) -> ok₁(sel₂(X1, X2))
indx₂(ok₁(X1), ok₁(X2)) -> ok₁(indx₂(X1, X2))
from₁(ok₁(X)) -> ok₁(from₁(X))
dbl1₁(ok₁(X)) -> ok₁(dbl1₁(X))
s1₁(ok₁(X)) -> ok₁(s1₁(X))
sel1₂(ok₁(X1), ok₁(X2)) -> ok₁(sel1₂(X1, X2))
quote₁(ok₁(X)) -> ok₁(quote₁(X))
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.