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 strictly oriented and are deleted.


FROM1(ok1(X)) -> FROM1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
FROM1(x1)  =  FROM1(x1)
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
ok1 > FROM1


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 strictly oriented and are deleted.


CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
CONS2(x1, x2)  =  CONS1(x1)
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


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 strictly oriented and are deleted.


S1(ok1(X)) -> S1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
S1(x1)  =  S1(x1)
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
ok1 > S1


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 strictly oriented and are deleted.


QUOTE1(ok1(X)) -> QUOTE1(X)
The remaining pairs can at least by weakly be oriented.

QUOTE1(mark1(X)) -> QUOTE1(X)
Used ordering: Combined order from the following AFS and order.
QUOTE1(x1)  =  QUOTE1(x1)
mark1(x1)  =  x1
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

QUOTE1(mark1(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 strictly oriented and are deleted.


QUOTE1(mark1(X)) -> QUOTE1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
QUOTE1(x1)  =  QUOTE1(x1)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
mark1 > QUOTE1


The following usable rules [14] were oriented: none



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

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 strictly oriented and are deleted.


SEL12(ok1(X1), ok1(X2)) -> SEL12(X1, X2)
The remaining pairs can at least by weakly be oriented.

SEL12(mark1(X1), X2) -> SEL12(X1, X2)
SEL12(X1, mark1(X2)) -> SEL12(X1, X2)
Used ordering: Combined order from the following AFS and order.
SEL12(x1, x2)  =  x2
mark1(x1)  =  x1
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


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)
SEL12(X1, mark1(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 strictly oriented and are deleted.


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

SEL12(X1, mark1(X2)) -> SEL12(X1, X2)
Used ordering: Combined order from the following AFS and order.
SEL12(x1, x2)  =  SEL11(x1)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
[SEL11, mark1]


The following usable rules [14] were oriented: none



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

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

SEL12(X1, mark1(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 strictly oriented and are deleted.


SEL12(X1, mark1(X2)) -> SEL12(X1, X2)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
SEL12(x1, x2)  =  SEL11(x2)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
[SEL11, mark1]


The following usable rules [14] were oriented: none



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

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

active1(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 strictly oriented and are deleted.


S11(ok1(X)) -> S11(X)
The remaining pairs can at least by weakly be oriented.

S11(mark1(X)) -> S11(X)
Used ordering: Combined order from the following AFS and order.
S11(x1)  =  S11(x1)
mark1(x1)  =  x1
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

S11(mark1(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 strictly oriented and are deleted.


S11(mark1(X)) -> S11(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
S11(x1)  =  S11(x1)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
mark1 > S11


The following usable rules [14] were oriented: none



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

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 strictly oriented and are deleted.


DBL11(mark1(X)) -> DBL11(X)
The remaining pairs can at least by weakly be oriented.

DBL11(ok1(X)) -> DBL11(X)
Used ordering: Combined order from the following AFS and order.
DBL11(x1)  =  DBL11(x1)
ok1(x1)  =  x1
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

DBL11(ok1(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 strictly oriented and are deleted.


DBL11(ok1(X)) -> DBL11(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
DBL11(x1)  =  DBL11(x1)
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
ok1 > DBL11


The following usable rules [14] were oriented: none



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

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

active1(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 strictly oriented and are deleted.


INDX2(ok1(X1), ok1(X2)) -> INDX2(X1, X2)
The remaining pairs can at least by weakly be oriented.

INDX2(mark1(X1), X2) -> INDX2(X1, X2)
Used ordering: Combined order from the following AFS and order.
INDX2(x1, x2)  =  INDX1(x2)
ok1(x1)  =  ok1(x1)
mark1(x1)  =  mark

Lexicographic Path Order [19].
Precedence:
ok1 > [INDX1, mark]


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 strictly oriented and are deleted.


INDX2(mark1(X1), X2) -> INDX2(X1, X2)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
INDX2(x1, x2)  =  INDX1(x1)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
[INDX1, mark1]


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 strictly oriented and are deleted.


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

SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
Used ordering: Combined order from the following AFS and order.
SEL2(x1, x2)  =  x2
mark1(x1)  =  x1
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


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)
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 strictly oriented and are deleted.


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

SEL2(mark1(X1), X2) -> SEL2(X1, X2)
Used ordering: Combined order from the following AFS and order.
SEL2(x1, x2)  =  SEL1(x2)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
[SEL1, mark1]


The following usable rules [14] were oriented: none



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

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

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

The TRS R consists of the following rules:

active1(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 strictly oriented and are deleted.


SEL2(mark1(X1), X2) -> SEL2(X1, X2)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
SEL2(x1, x2)  =  SEL1(x1)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
[SEL1, mark1]


The following usable rules [14] were oriented: none



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

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 strictly oriented and are deleted.


DBLS1(mark1(X)) -> DBLS1(X)
The remaining pairs can at least by weakly be oriented.

DBLS1(ok1(X)) -> DBLS1(X)
Used ordering: Combined order from the following AFS and order.
DBLS1(x1)  =  DBLS1(x1)
ok1(x1)  =  x1
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

DBLS1(ok1(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 strictly oriented and are deleted.


DBLS1(ok1(X)) -> DBLS1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
DBLS1(x1)  =  DBLS1(x1)
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
ok1 > DBLS1


The following usable rules [14] were oriented: none



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

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 strictly oriented and are deleted.


DBL1(ok1(X)) -> DBL1(X)
The remaining pairs can at least by weakly be oriented.

DBL1(mark1(X)) -> DBL1(X)
Used ordering: Combined order from the following AFS and order.
DBL1(x1)  =  DBL1(x1)
mark1(x1)  =  x1
ok1(x1)  =  ok1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


The following usable rules [14] were oriented: none



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

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

DBL1(mark1(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 strictly oriented and are deleted.


DBL1(mark1(X)) -> DBL1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
DBL1(x1)  =  DBL1(x1)
mark1(x1)  =  mark1(x1)

Lexicographic Path Order [19].
Precedence:
mark1 > DBL1


The following usable rules [14] were oriented: none



↳ QTRS
  ↳ DependencyPairsProof
    ↳ QDP
      ↳ DependencyGraphProof
        ↳ AND
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
          ↳ QDP
            ↳ QDPOrderProof
              ↳ QDP
                ↳ QDPOrderProof
QDP
                    ↳ 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 strictly oriented and are deleted.


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(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)
The remaining pairs can at least by weakly be oriented.

PROPER1(quote1(X)) -> PROPER1(X)
PROPER1(from1(X)) -> PROPER1(X)
PROPER1(dbl1(X)) -> PROPER1(X)
Used ordering: Combined order from the following AFS and order.
PROPER1(x1)  =  PROPER1(x1)
quote1(x1)  =  x1
cons2(x1, x2)  =  cons2(x1, x2)
sel2(x1, x2)  =  sel2(x1, x2)
s11(x1)  =  s11(x1)
indx2(x1, x2)  =  indx2(x1, x2)
sel12(x1, x2)  =  sel12(x1, x2)
s1(x1)  =  s1(x1)
from1(x1)  =  x1
dbls1(x1)  =  dbls1(x1)
dbl11(x1)  =  dbl11(x1)
dbl1(x1)  =  x1

Lexicographic Path Order [19].
Precedence:
cons2 > [PROPER1, dbl11]
indx2 > [PROPER1, dbl11]


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(quote1(X)) -> PROPER1(X)
PROPER1(from1(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 strictly oriented and are deleted.


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

PROPER1(quote1(X)) -> PROPER1(X)
PROPER1(dbl1(X)) -> PROPER1(X)
Used ordering: Combined order from the following AFS and order.
PROPER1(x1)  =  PROPER1(x1)
quote1(x1)  =  x1
from1(x1)  =  from1(x1)
dbl1(x1)  =  x1

Lexicographic Path Order [19].
Precedence:
[PROPER1, from1]


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
                    ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(quote1(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 strictly oriented and are deleted.


PROPER1(dbl1(X)) -> PROPER1(X)
The remaining pairs can at least by weakly be oriented.

PROPER1(quote1(X)) -> PROPER1(X)
Used ordering: Combined order from the following AFS and order.
PROPER1(x1)  =  PROPER1(x1)
quote1(x1)  =  x1
dbl1(x1)  =  dbl1(x1)

Lexicographic Path Order [19].
Precedence:
trivial


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
                    ↳ QDPOrderProof
QDP
                        ↳ QDPOrderProof
          ↳ QDP
          ↳ QDP

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

PROPER1(quote1(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 strictly oriented and are deleted.


PROPER1(quote1(X)) -> PROPER1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
PROPER1(x1)  =  PROPER1(x1)
quote1(x1)  =  quote1(x1)

Lexicographic Path Order [19].
Precedence:
quote1 > PROPER1


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
                    ↳ 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 strictly oriented and are deleted.


ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(dbl1(X)) -> ACTIVE1(X)
ACTIVE1(sel12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(dbls1(X)) -> ACTIVE1(X)
The remaining pairs can at least by weakly be oriented.

ACTIVE1(dbl11(X)) -> ACTIVE1(X)
ACTIVE1(quote1(X)) -> ACTIVE1(X)
ACTIVE1(indx2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s11(X)) -> ACTIVE1(X)
Used ordering: Combined order from the following AFS and order.
ACTIVE1(x1)  =  ACTIVE1(x1)
sel12(x1, x2)  =  sel12(x1, x2)
dbl11(x1)  =  x1
sel2(x1, x2)  =  sel2(x1, x2)
quote1(x1)  =  x1
indx2(x1, x2)  =  x1
dbl1(x1)  =  dbl1(x1)
dbls1(x1)  =  dbls1(x1)
s11(x1)  =  x1

Lexicographic Path Order [19].
Precedence:
dbl1 > [ACTIVE1, sel12, dbls1]


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(dbl11(X)) -> ACTIVE1(X)
ACTIVE1(quote1(X)) -> ACTIVE1(X)
ACTIVE1(indx2(X1, X2)) -> ACTIVE1(X1)
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 strictly oriented and are deleted.


ACTIVE1(quote1(X)) -> ACTIVE1(X)
The remaining pairs can at least by weakly be oriented.

ACTIVE1(dbl11(X)) -> ACTIVE1(X)
ACTIVE1(indx2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(s11(X)) -> ACTIVE1(X)
Used ordering: Combined order from the following AFS and order.
ACTIVE1(x1)  =  ACTIVE1(x1)
dbl11(x1)  =  x1
quote1(x1)  =  quote1(x1)
indx2(x1, x2)  =  x1
s11(x1)  =  x1

Lexicographic Path Order [19].
Precedence:
trivial


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
                    ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(dbl11(X)) -> ACTIVE1(X)
ACTIVE1(indx2(X1, X2)) -> ACTIVE1(X1)
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 strictly oriented and are deleted.


ACTIVE1(indx2(X1, X2)) -> ACTIVE1(X1)
The remaining pairs can at least by weakly be oriented.

ACTIVE1(dbl11(X)) -> ACTIVE1(X)
ACTIVE1(s11(X)) -> ACTIVE1(X)
Used ordering: Combined order from the following AFS and order.
ACTIVE1(x1)  =  ACTIVE1(x1)
dbl11(x1)  =  x1
indx2(x1, x2)  =  indx2(x1, x2)
s11(x1)  =  x1

Lexicographic Path Order [19].
Precedence:
trivial


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
                    ↳ QDPOrderProof
QDP
                        ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(dbl11(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 strictly oriented and are deleted.


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

ACTIVE1(dbl11(X)) -> ACTIVE1(X)
Used ordering: Combined order from the following AFS and order.
ACTIVE1(x1)  =  ACTIVE1(x1)
dbl11(x1)  =  x1
s11(x1)  =  s11(x1)

Lexicographic Path Order [19].
Precedence:
trivial


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
                    ↳ QDPOrderProof
                      ↳ QDP
                        ↳ QDPOrderProof
QDP
                            ↳ QDPOrderProof
          ↳ QDP

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

ACTIVE1(dbl11(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 strictly oriented and are deleted.


ACTIVE1(dbl11(X)) -> ACTIVE1(X)
The remaining pairs can at least by weakly be oriented.
none
Used ordering: Combined order from the following AFS and order.
ACTIVE1(x1)  =  ACTIVE1(x1)
dbl11(x1)  =  dbl11(x1)

Lexicographic Path Order [19].
Precedence:
dbl11 > ACTIVE1


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
                    ↳ QDPOrderProof
                      ↳ 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.