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(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.


QTRS
  ↳ DependencyPairsProof

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

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.

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

ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fib1(N)) -> SEL2(N, fib12(s1(0), s1(0)))
PROPER1(fib12(X1, X2)) -> FIB12(proper1(X1), proper1(X2))
ACTIVE1(fib12(X1, X2)) -> FIB12(X1, active1(X2))
S1(mark1(X)) -> S1(X)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fib12(X, Y)) -> FIB12(Y, add2(X, Y))
FIB1(mark1(X)) -> FIB1(X)
TOP1(mark1(X)) -> TOP1(proper1(X))
ACTIVE1(fib1(N)) -> S1(0)
FIB12(mark1(X1), X2) -> FIB12(X1, X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(add2(X1, X2)) -> ADD2(active1(X1), X2)
PROPER1(fib12(X1, X2)) -> PROPER1(X2)
ACTIVE1(fib1(N)) -> FIB12(s1(0), s1(0))
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
TOP1(ok1(X)) -> ACTIVE1(X)
FIB1(ok1(X)) -> FIB1(X)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
ACTIVE1(s1(X)) -> S1(active1(X))
ACTIVE1(fib12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(add2(X1, X2)) -> ADD2(X1, active1(X2))
FIB12(X1, mark1(X2)) -> FIB12(X1, X2)
PROPER1(s1(X)) -> PROPER1(X)
ACTIVE1(add2(s1(X), Y)) -> ADD2(X, Y)
TOP1(mark1(X)) -> PROPER1(X)
ADD2(mark1(X1), X2) -> ADD2(X1, X2)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X1)
PROPER1(fib1(X)) -> PROPER1(X)
ACTIVE1(fib1(X)) -> ACTIVE1(X)
PROPER1(sel2(X1, X2)) -> SEL2(proper1(X1), proper1(X2))
ACTIVE1(fib12(X1, X2)) -> ACTIVE1(X1)
PROPER1(add2(X1, X2)) -> PROPER1(X1)
ACTIVE1(sel2(X1, X2)) -> SEL2(X1, active1(X2))
PROPER1(add2(X1, X2)) -> ADD2(proper1(X1), proper1(X2))
ACTIVE1(fib12(X1, X2)) -> FIB12(active1(X1), X2)
ACTIVE1(sel2(s1(N), cons2(X, XS))) -> SEL2(N, XS)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
PROPER1(s1(X)) -> S1(proper1(X))
S1(ok1(X)) -> S1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ADD2(ok1(X1), ok1(X2)) -> ADD2(X1, X2)
PROPER1(fib1(X)) -> FIB1(proper1(X))
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fib1(X)) -> FIB1(active1(X))
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(fib12(X1, X2)) -> PROPER1(X1)
ACTIVE1(fib12(X, Y)) -> ADD2(X, Y)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
ACTIVE1(fib12(X, Y)) -> CONS2(X, fib12(Y, add2(X, Y)))
FIB12(ok1(X1), ok1(X2)) -> FIB12(X1, X2)
PROPER1(add2(X1, X2)) -> PROPER1(X2)
ADD2(X1, mark1(X2)) -> ADD2(X1, X2)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
TOP1(ok1(X)) -> TOP1(active1(X))
ACTIVE1(sel2(X1, X2)) -> SEL2(active1(X1), X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(add2(s1(X), Y)) -> S1(add2(X, Y))

The TRS R consists of the following rules:

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

↳ QTRS
  ↳ DependencyPairsProof
QDP
      ↳ DependencyGraphProof

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

ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fib1(N)) -> SEL2(N, fib12(s1(0), s1(0)))
PROPER1(fib12(X1, X2)) -> FIB12(proper1(X1), proper1(X2))
ACTIVE1(fib12(X1, X2)) -> FIB12(X1, active1(X2))
S1(mark1(X)) -> S1(X)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(fib12(X, Y)) -> FIB12(Y, add2(X, Y))
FIB1(mark1(X)) -> FIB1(X)
TOP1(mark1(X)) -> TOP1(proper1(X))
ACTIVE1(fib1(N)) -> S1(0)
FIB12(mark1(X1), X2) -> FIB12(X1, X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
ACTIVE1(add2(X1, X2)) -> ADD2(active1(X1), X2)
PROPER1(fib12(X1, X2)) -> PROPER1(X2)
ACTIVE1(fib1(N)) -> FIB12(s1(0), s1(0))
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
TOP1(ok1(X)) -> ACTIVE1(X)
FIB1(ok1(X)) -> FIB1(X)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
ACTIVE1(s1(X)) -> S1(active1(X))
ACTIVE1(fib12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(add2(X1, X2)) -> ADD2(X1, active1(X2))
FIB12(X1, mark1(X2)) -> FIB12(X1, X2)
PROPER1(s1(X)) -> PROPER1(X)
ACTIVE1(add2(s1(X), Y)) -> ADD2(X, Y)
TOP1(mark1(X)) -> PROPER1(X)
ADD2(mark1(X1), X2) -> ADD2(X1, X2)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X1)
PROPER1(fib1(X)) -> PROPER1(X)
ACTIVE1(fib1(X)) -> ACTIVE1(X)
PROPER1(sel2(X1, X2)) -> SEL2(proper1(X1), proper1(X2))
ACTIVE1(fib12(X1, X2)) -> ACTIVE1(X1)
PROPER1(add2(X1, X2)) -> PROPER1(X1)
ACTIVE1(sel2(X1, X2)) -> SEL2(X1, active1(X2))
PROPER1(add2(X1, X2)) -> ADD2(proper1(X1), proper1(X2))
ACTIVE1(fib12(X1, X2)) -> FIB12(active1(X1), X2)
ACTIVE1(sel2(s1(N), cons2(X, XS))) -> SEL2(N, XS)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
PROPER1(s1(X)) -> S1(proper1(X))
S1(ok1(X)) -> S1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
CONS2(mark1(X1), X2) -> CONS2(X1, X2)
ACTIVE1(cons2(X1, X2)) -> CONS2(active1(X1), X2)
ADD2(ok1(X1), ok1(X2)) -> ADD2(X1, X2)
PROPER1(fib1(X)) -> FIB1(proper1(X))
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fib1(X)) -> FIB1(active1(X))
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(fib12(X1, X2)) -> PROPER1(X1)
ACTIVE1(fib12(X, Y)) -> ADD2(X, Y)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
ACTIVE1(fib12(X, Y)) -> CONS2(X, fib12(Y, add2(X, Y)))
FIB12(ok1(X1), ok1(X2)) -> FIB12(X1, X2)
PROPER1(add2(X1, X2)) -> PROPER1(X2)
ADD2(X1, mark1(X2)) -> ADD2(X1, X2)
PROPER1(cons2(X1, X2)) -> CONS2(proper1(X1), proper1(X2))
TOP1(ok1(X)) -> TOP1(active1(X))
ACTIVE1(sel2(X1, X2)) -> SEL2(active1(X1), X2)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
ACTIVE1(add2(s1(X), Y)) -> S1(add2(X, Y))

The TRS R consists of the following rules:

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 9 SCCs with 26 less nodes.

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

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

ADD2(X1, mark1(X2)) -> ADD2(X1, X2)
ADD2(mark1(X1), X2) -> ADD2(X1, X2)
ADD2(ok1(X1), ok1(X2)) -> ADD2(X1, X2)

The TRS R consists of the following rules:

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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


The following pairs can be oriented strictly and are deleted.


ADD2(X1, mark1(X2)) -> ADD2(X1, X2)
ADD2(mark1(X1), X2) -> ADD2(X1, X2)
ADD2(ok1(X1), ok1(X2)) -> ADD2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ADD2(x1, x2)) = 3·x1 + 3·x1·x2 + 3·x2   
POL(mark1(x1)) = 3 + x1   
POL(ok1(x1)) = 3 + 2·x1   

The following usable rules [14] were oriented: none



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

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

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

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

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

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

The TRS R consists of the following rules:

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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


The following pairs can be oriented strictly and are deleted.


CONS2(mark1(X1), X2) -> CONS2(X1, X2)
CONS2(ok1(X1), ok1(X2)) -> CONS2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(CONS2(x1, x2)) = 3·x1 + 3·x1·x2 + 3·x2   
POL(mark1(x1)) = 3 + x1   
POL(ok1(x1)) = 3 + 2·x1   

The following usable rules [14] were oriented: none



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

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

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

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

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

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

The TRS R consists of the following rules:

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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


The following pairs can be oriented strictly and are deleted.


S1(ok1(X)) -> S1(X)
S1(mark1(X)) -> S1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(S1(x1)) = 3·x1 + 3·x12   
POL(mark1(x1)) = 3 + 3·x1 + 3·x12   
POL(ok1(x1)) = 3 + x1 + 2·x12   

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

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

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

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

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

FIB12(X1, mark1(X2)) -> FIB12(X1, X2)
FIB12(mark1(X1), X2) -> FIB12(X1, X2)
FIB12(ok1(X1), ok1(X2)) -> FIB12(X1, X2)

The TRS R consists of the following rules:

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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


The following pairs can be oriented strictly and are deleted.


FIB12(X1, mark1(X2)) -> FIB12(X1, X2)
FIB12(mark1(X1), X2) -> FIB12(X1, X2)
FIB12(ok1(X1), ok1(X2)) -> FIB12(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(FIB12(x1, x2)) = 3·x1 + 3·x1·x2 + 3·x2   
POL(mark1(x1)) = 3 + x1   
POL(ok1(x1)) = 3 + 2·x1   

The following usable rules [14] were oriented: none



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

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

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

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

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(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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


The following pairs can be oriented strictly and are deleted.


SEL2(X1, mark1(X2)) -> SEL2(X1, X2)
SEL2(ok1(X1), ok1(X2)) -> SEL2(X1, X2)
SEL2(mark1(X1), X2) -> SEL2(X1, X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(SEL2(x1, x2)) = 3·x1 + 3·x1·x2 + 3·x2   
POL(mark1(x1)) = 3 + x1   
POL(ok1(x1)) = 3 + 2·x1   

The following usable rules [14] were oriented: none



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

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

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

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

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

FIB1(mark1(X)) -> FIB1(X)
FIB1(ok1(X)) -> FIB1(X)

The TRS R consists of the following rules:

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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


The following pairs can be oriented strictly and are deleted.


FIB1(mark1(X)) -> FIB1(X)
FIB1(ok1(X)) -> FIB1(X)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(FIB1(x1)) = 3·x1 + 3·x12   
POL(mark1(x1)) = 3 + x1 + 2·x12   
POL(ok1(x1)) = 3 + 3·x1 + 3·x12   

The following usable rules [14] were oriented: none



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

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

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

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

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

PROPER1(add2(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(fib12(X1, X2)) -> PROPER1(X1)
PROPER1(fib1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(fib12(X1, X2)) -> PROPER1(X2)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
PROPER1(add2(X1, X2)) -> PROPER1(X2)

The TRS R consists of the following rules:

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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


The following pairs can be oriented strictly and are deleted.


PROPER1(add2(X1, X2)) -> PROPER1(X1)
PROPER1(s1(X)) -> PROPER1(X)
PROPER1(cons2(X1, X2)) -> PROPER1(X1)
PROPER1(fib12(X1, X2)) -> PROPER1(X1)
PROPER1(fib1(X)) -> PROPER1(X)
PROPER1(sel2(X1, X2)) -> PROPER1(X1)
PROPER1(cons2(X1, X2)) -> PROPER1(X2)
PROPER1(fib12(X1, X2)) -> PROPER1(X2)
PROPER1(sel2(X1, X2)) -> PROPER1(X2)
PROPER1(add2(X1, X2)) -> PROPER1(X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(PROPER1(x1)) = 3·x1 + 3·x12   
POL(add2(x1, x2)) = 3 + x1 + 2·x2   
POL(cons2(x1, x2)) = 3 + x1 + 2·x2   
POL(fib1(x1)) = 3 + 3·x1 + 3·x12   
POL(fib12(x1, x2)) = 3 + 2·x1 + 2·x2   
POL(s1(x1)) = 3 + 2·x1 + 2·x12   
POL(sel2(x1, x2)) = 3 + 2·x1 + 2·x2   

The following usable rules [14] were oriented: none



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

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

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

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

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

ACTIVE1(fib12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fib12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fib1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X2)

The TRS R consists of the following rules:

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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


The following pairs can be oriented strictly and are deleted.


ACTIVE1(fib12(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(cons2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fib12(X1, X2)) -> ACTIVE1(X2)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(sel2(X1, X2)) -> ACTIVE1(X1)
ACTIVE1(fib1(X)) -> ACTIVE1(X)
ACTIVE1(s1(X)) -> ACTIVE1(X)
ACTIVE1(add2(X1, X2)) -> ACTIVE1(X2)
The remaining pairs can at least be oriented weakly.
none
Used ordering: Polynomial interpretation [21]:

POL(ACTIVE1(x1)) = 3·x1 + 3·x12   
POL(add2(x1, x2)) = 3 + 2·x1 + 2·x2   
POL(cons2(x1, x2)) = 3 + x1   
POL(fib1(x1)) = 3 + 3·x1 + 3·x12   
POL(fib12(x1, x2)) = 3 + x1 + x2   
POL(s1(x1)) = 3 + 3·x1 + 3·x12   
POL(sel2(x1, x2)) = 3 + 2·x1 + x2   

The following usable rules [14] were oriented: none



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

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

active1(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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

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

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(fib1(N)) -> mark1(sel2(N, fib12(s1(0), s1(0))))
active1(fib12(X, Y)) -> mark1(cons2(X, fib12(Y, add2(X, Y))))
active1(add2(0, X)) -> mark1(X)
active1(add2(s1(X), Y)) -> mark1(s1(add2(X, Y)))
active1(sel2(0, cons2(X, XS))) -> mark1(X)
active1(sel2(s1(N), cons2(X, XS))) -> mark1(sel2(N, XS))
active1(fib1(X)) -> fib1(active1(X))
active1(sel2(X1, X2)) -> sel2(active1(X1), X2)
active1(sel2(X1, X2)) -> sel2(X1, active1(X2))
active1(fib12(X1, X2)) -> fib12(active1(X1), X2)
active1(fib12(X1, X2)) -> fib12(X1, active1(X2))
active1(s1(X)) -> s1(active1(X))
active1(cons2(X1, X2)) -> cons2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(active1(X1), X2)
active1(add2(X1, X2)) -> add2(X1, active1(X2))
fib1(mark1(X)) -> mark1(fib1(X))
sel2(mark1(X1), X2) -> mark1(sel2(X1, X2))
sel2(X1, mark1(X2)) -> mark1(sel2(X1, X2))
fib12(mark1(X1), X2) -> mark1(fib12(X1, X2))
fib12(X1, mark1(X2)) -> mark1(fib12(X1, X2))
s1(mark1(X)) -> mark1(s1(X))
cons2(mark1(X1), X2) -> mark1(cons2(X1, X2))
add2(mark1(X1), X2) -> mark1(add2(X1, X2))
add2(X1, mark1(X2)) -> mark1(add2(X1, X2))
proper1(fib1(X)) -> fib1(proper1(X))
proper1(sel2(X1, X2)) -> sel2(proper1(X1), proper1(X2))
proper1(fib12(X1, X2)) -> fib12(proper1(X1), proper1(X2))
proper1(s1(X)) -> s1(proper1(X))
proper1(0) -> ok1(0)
proper1(cons2(X1, X2)) -> cons2(proper1(X1), proper1(X2))
proper1(add2(X1, X2)) -> add2(proper1(X1), proper1(X2))
fib1(ok1(X)) -> ok1(fib1(X))
sel2(ok1(X1), ok1(X2)) -> ok1(sel2(X1, X2))
fib12(ok1(X1), ok1(X2)) -> ok1(fib12(X1, X2))
s1(ok1(X)) -> ok1(s1(X))
cons2(ok1(X1), ok1(X2)) -> ok1(cons2(X1, X2))
add2(ok1(X1), ok1(X2)) -> ok1(add2(X1, X2))
top1(mark1(X)) -> top1(proper1(X))
top1(ok1(X)) -> top1(active1(X))

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