Termination w.r.t. Q proof of /tmp/tmp9VLwQP/Ex8_BLR02

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:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Q is empty.

Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

ADD(active(X1), X2) → ADD(X1, X2)
MARK(fib1(X1, X2)) → MARK(X1)
ACTIVE(fib(N)) → MARK(sel(N, fib1(s(0), s(0))))
MARK(fib1(X1, X2)) → FIB1(mark(X1), mark(X2))
MARK(fib(X)) → ACTIVE(fib(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
ADD(X1, active(X2)) → ADD(X1, X2)
ADD(mark(X1), X2) → ADD(X1, X2)
ACTIVE(fib1(X, Y)) → CONS(X, fib1(Y, add(X, Y)))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(add(X1, X2)) → ACTIVE(add(mark(X1), mark(X2)))
FIB1(X1, active(X2)) → FIB1(X1, X2)
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
FIB(active(X)) → FIB(X)
FIB1(X1, mark(X2)) → FIB1(X1, X2)
FIB1(active(X1), X2) → FIB1(X1, X2)
MARK(add(X1, X2)) → ADD(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
S(active(X)) → S(X)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
FIB(mark(X)) → FIB(X)
MARK(fib(X)) → FIB(mark(X))
ACTIVE(fib(N)) → FIB1(s(0), s(0))
ACTIVE(fib1(X, Y)) → FIB1(Y, add(X, Y))
ACTIVE(fib(N)) → SEL(N, fib1(s(0), s(0)))
ACTIVE(fib(N)) → S(0)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(fib1(X, Y)) → ADD(X, Y)
CONS(mark(X1), X2) → CONS(X1, X2)
MARK(fib(X)) → MARK(X)
SEL(mark(X1), X2) → SEL(X1, X2)
ACTIVE(add(s(X), Y)) → ADD(X, Y)
MARK(s(X)) → MARK(X)
SEL(X1, active(X2)) → SEL(X1, X2)
ACTIVE(add(s(X), Y)) → S(add(X, Y))
MARK(add(X1, X2)) → MARK(X2)
ACTIVE(add(s(X), Y)) → MARK(s(add(X, Y)))
MARK(fib1(X1, X2)) → MARK(X2)
ADD(X1, mark(X2)) → ADD(X1, X2)
MARK(add(X1, X2)) → MARK(X1)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
FIB1(mark(X1), X2) → FIB1(X1, X2)
MARK(s(X)) → S(mark(X))
S(mark(X)) → S(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(fib1(X, Y)) → MARK(cons(X, fib1(Y, add(X, Y))))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
MARK(fib1(X1, X2)) → ACTIVE(fib1(mark(X1), mark(X2)))
MARK(0) → ACTIVE(0)
ACTIVE(sel(s(N), cons(X, XS))) → SEL(N, XS)
ACTIVE(add(0, X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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:

ADD(active(X1), X2) → ADD(X1, X2)
MARK(fib1(X1, X2)) → MARK(X1)
ACTIVE(fib(N)) → MARK(sel(N, fib1(s(0), s(0))))
MARK(fib1(X1, X2)) → FIB1(mark(X1), mark(X2))
MARK(fib(X)) → ACTIVE(fib(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
CONS(X1, mark(X2)) → CONS(X1, X2)
ADD(X1, active(X2)) → ADD(X1, X2)
ADD(mark(X1), X2) → ADD(X1, X2)
ACTIVE(fib1(X, Y)) → CONS(X, fib1(Y, add(X, Y)))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(add(X1, X2)) → ACTIVE(add(mark(X1), mark(X2)))
FIB1(X1, active(X2)) → FIB1(X1, X2)
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
FIB(active(X)) → FIB(X)
FIB1(X1, mark(X2)) → FIB1(X1, X2)
FIB1(active(X1), X2) → FIB1(X1, X2)
MARK(add(X1, X2)) → ADD(mark(X1), mark(X2))
MARK(sel(X1, X2)) → MARK(X1)
S(active(X)) → S(X)
MARK(cons(X1, X2)) → CONS(mark(X1), X2)
FIB(mark(X)) → FIB(X)
MARK(fib(X)) → FIB(mark(X))
ACTIVE(fib(N)) → FIB1(s(0), s(0))
ACTIVE(fib1(X, Y)) → FIB1(Y, add(X, Y))
ACTIVE(fib(N)) → SEL(N, fib1(s(0), s(0)))
ACTIVE(fib(N)) → S(0)
CONS(active(X1), X2) → CONS(X1, X2)
ACTIVE(fib1(X, Y)) → ADD(X, Y)
CONS(mark(X1), X2) → CONS(X1, X2)
MARK(fib(X)) → MARK(X)
SEL(mark(X1), X2) → SEL(X1, X2)
ACTIVE(add(s(X), Y)) → ADD(X, Y)
MARK(s(X)) → MARK(X)
SEL(X1, active(X2)) → SEL(X1, X2)
ACTIVE(add(s(X), Y)) → S(add(X, Y))
MARK(add(X1, X2)) → MARK(X2)
ACTIVE(add(s(X), Y)) → MARK(s(add(X, Y)))
MARK(fib1(X1, X2)) → MARK(X2)
ADD(X1, mark(X2)) → ADD(X1, X2)
MARK(add(X1, X2)) → MARK(X1)
CONS(X1, active(X2)) → CONS(X1, X2)
MARK(s(X)) → ACTIVE(s(mark(X)))
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)
FIB1(mark(X1), X2) → FIB1(X1, X2)
MARK(s(X)) → S(mark(X))
S(mark(X)) → S(X)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
ACTIVE(fib1(X, Y)) → MARK(cons(X, fib1(Y, add(X, Y))))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(sel(X1, X2)) → SEL(mark(X1), mark(X2))
MARK(fib1(X1, X2)) → ACTIVE(fib1(mark(X1), mark(X2)))
MARK(0) → ACTIVE(0)
ACTIVE(sel(s(N), cons(X, XS))) → SEL(N, XS)
ACTIVE(add(0, X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 7 SCCs with 16 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ADD(active(X1), X2) → ADD(X1, X2)
ADD(X1, active(X2)) → ADD(X1, X2)
ADD(X1, mark(X2)) → ADD(X1, X2)
ADD(mark(X1), X2) → ADD(X1, X2)

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We can use the usable rules and reduction pair processor [15] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its argument. Then, we can delete all non-usable rules [17] from R.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

ADD(active(X1), X2) → ADD(X1, X2)
ADD(X1, active(X2)) → ADD(X1, X2)
ADD(X1, mark(X2)) → ADD(X1, X2)
ADD(mark(X1), X2) → ADD(X1, X2)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

ADD(active(X1), X2) → ADD(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
ADD(X1, active(X2)) → ADD(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
ADD(X1, mark(X2)) → ADD(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
ADD(mark(X1), X2) → ADD(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

CONS(X1, active(X2)) → CONS(X1, X2)
CONS(mark(X1), X2) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

CONS(mark(X1), X2) → CONS(X1, X2)
CONS(X1, active(X2)) → CONS(X1, X2)
CONS(X1, mark(X2)) → CONS(X1, X2)
CONS(active(X1), X2) → CONS(X1, X2)

From the DPs we obtained the following set of size-change graphs:

CONS(X1, active(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
CONS(mark(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(active(X1), X2) → CONS(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
CONS(X1, mark(X2)) → CONS(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(mark(X)) → S(X)
S(active(X)) → S(X)

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

S(active(X)) → S(X)
S(mark(X)) → S(X)

From the DPs we obtained the following set of size-change graphs:

S(mark(X)) → S(X)
The graph contains the following edges 1 > 1
S(active(X)) → S(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

FIB1(X1, active(X2)) → FIB1(X1, X2)
FIB1(X1, mark(X2)) → FIB1(X1, X2)
FIB1(mark(X1), X2) → FIB1(X1, X2)
FIB1(active(X1), X2) → FIB1(X1, X2)

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

          ↳ QDP

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

FIB1(X1, active(X2)) → FIB1(X1, X2)
FIB1(X1, mark(X2)) → FIB1(X1, X2)
FIB1(mark(X1), X2) → FIB1(X1, X2)
FIB1(active(X1), X2) → FIB1(X1, X2)

From the DPs we obtained the following set of size-change graphs:

FIB1(X1, active(X2)) → FIB1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
FIB1(X1, mark(X2)) → FIB1(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
FIB1(mark(X1), X2) → FIB1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
FIB1(active(X1), X2) → FIB1(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

          ↳ QDP

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

SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

          ↳ QDP

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

SEL(mark(X1), X2) → SEL(X1, X2)
SEL(X1, active(X2)) → SEL(X1, X2)
SEL(X1, mark(X2)) → SEL(X1, X2)
SEL(active(X1), X2) → SEL(X1, X2)

From the DPs we obtained the following set of size-change graphs:

SEL(mark(X1), X2) → SEL(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2
SEL(X1, active(X2)) → SEL(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL(X1, mark(X2)) → SEL(X1, X2)
The graph contains the following edges 1 >= 1, 2 > 2
SEL(active(X1), X2) → SEL(X1, X2)
The graph contains the following edges 1 > 1, 2 >= 2

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

FIB(mark(X)) → FIB(X)
FIB(active(X)) → FIB(X)

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ UsableRulesProof

              ↳ QDP

                ↳ QDPSizeChangeProof

          ↳ QDP

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

FIB(mark(X)) → FIB(X)
FIB(active(X)) → FIB(X)

From the DPs we obtained the following set of size-change graphs:

FIB(mark(X)) → FIB(X)
The graph contains the following edges 1 > 1
FIB(active(X)) → FIB(X)
The graph contains the following edges 1 > 1

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

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

MARK(fib(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
ACTIVE(fib(N)) → MARK(sel(N, fib1(s(0), s(0))))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → ACTIVE(fib(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(fib1(X, Y)) → MARK(cons(X, fib1(Y, add(X, Y))))
MARK(fib1(X1, X2)) → MARK(X2)
ACTIVE(add(s(X), Y)) → MARK(s(add(X, Y)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(add(X1, X2)) → MARK(X1)
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(add(X1, X2)) → ACTIVE(add(mark(X1), mark(X2)))
MARK(s(X)) → ACTIVE(s(mark(X)))
MARK(fib1(X1, X2)) → ACTIVE(fib1(mark(X1), mark(X2)))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(add(0, X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].

The following pairs can be oriented strictly and are deleted.

MARK(cons(X1, X2)) → ACTIVE(cons(mark(X1), X2))
MARK(s(X)) → ACTIVE(s(mark(X)))

The remaining pairs can at least be oriented weakly.

MARK(fib(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
ACTIVE(fib(N)) → MARK(sel(N, fib1(s(0), s(0))))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → ACTIVE(fib(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(fib1(X, Y)) → MARK(cons(X, fib1(Y, add(X, Y))))
MARK(fib1(X1, X2)) → MARK(X2)
ACTIVE(add(s(X), Y)) → MARK(s(add(X, Y)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(add(X1, X2)) → MARK(X1)
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(add(X1, X2)) → ACTIVE(add(mark(X1), mark(X2)))
MARK(fib1(X1, X2)) → ACTIVE(fib1(mark(X1), mark(X2)))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(add(0, X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

Used ordering: Polynomial interpretation [25]:

POL(0) = 0
POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1
POL(active(x₁)) = 0
POL(add(x₁, x₂)) = 1
POL(cons(x₁, x₂)) = 0
POL(fib(x₁)) = 1
POL(fib1(x₁, x₂)) = 1
POL(mark(x₁)) = 0
POL(s(x₁)) = 0
POL(sel(x₁, x₂)) = 1

The following usable rules [17] were oriented:

add(X1, mark(X2)) → add(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
s(active(X)) → s(X)
s(mark(X)) → s(X)
fib(active(X)) → fib(X)
fib(mark(X)) → fib(X)
fib1(X1, active(X2)) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

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

MARK(fib(X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X1)
ACTIVE(fib(N)) → MARK(sel(N, fib1(s(0), s(0))))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(fib(X)) → ACTIVE(fib(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
ACTIVE(fib1(X, Y)) → MARK(cons(X, fib1(Y, add(X, Y))))
MARK(fib1(X1, X2)) → MARK(X2)
ACTIVE(add(s(X), Y)) → MARK(s(add(X, Y)))
MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2)))
MARK(add(X1, X2)) → MARK(X1)
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(add(X1, X2)) → ACTIVE(add(mark(X1), mark(X2)))
MARK(fib1(X1, X2)) → ACTIVE(fib1(mark(X1), mark(X2)))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
ACTIVE(add(0, X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(sel(X1, X2)) → ACTIVE(sel(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(sel(fib1(x0, x1), y1)) → ACTIVE(sel(active(fib1(mark(x0), mark(x1))), mark(y1)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel(add(x0, x1), y1)) → ACTIVE(sel(active(add(mark(x0), mark(x1))), mark(y1)))
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(sel(y0, fib(x0))) → ACTIVE(sel(mark(y0), active(fib(mark(x0)))))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel(y0, fib1(x0, x1))) → ACTIVE(sel(mark(y0), active(fib1(mark(x0), mark(x1)))))
MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(mark(x0), x1))))
MARK(sel(fib(x0), y1)) → ACTIVE(sel(active(fib(mark(x0))), mark(y1)))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(mark(x0)))))
MARK(sel(y0, add(x0, x1))) → ACTIVE(sel(mark(y0), active(add(mark(x0), mark(x1)))))
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(mark(x0))), mark(y1)))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

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

MARK(fib(X)) → MARK(X)
ACTIVE(fib(N)) → MARK(sel(N, fib1(s(0), s(0))))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(fib1(x0, x1), y1)) → ACTIVE(sel(active(fib1(mark(x0), mark(x1))), mark(y1)))
MARK(fib(X)) → ACTIVE(fib(mark(X)))
MARK(cons(X1, X2)) → MARK(X1)
MARK(fib1(X1, X2)) → MARK(X2)
ACTIVE(add(s(X), Y)) → MARK(s(add(X, Y)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel(add(x0, x1), y1)) → ACTIVE(sel(active(add(mark(x0), mark(x1))), mark(y1)))
MARK(add(X1, X2)) → MARK(X1)
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(sel(y0, fib1(x0, x1))) → ACTIVE(sel(mark(y0), active(fib1(mark(x0), mark(x1)))))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(sel(fib(x0), y1)) → ACTIVE(sel(active(fib(mark(x0))), mark(y1)))
MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(mark(x0), x1))))
MARK(add(X1, X2)) → ACTIVE(add(mark(X1), mark(X2)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(mark(x0)))))
MARK(sel(y0, add(x0, x1))) → ACTIVE(sel(mark(y0), active(add(mark(x0), mark(x1)))))
MARK(sel(X1, X2)) → MARK(X1)
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
ACTIVE(fib1(X, Y)) → MARK(cons(X, fib1(Y, add(X, Y))))
MARK(sel(y0, fib(x0))) → ACTIVE(sel(mark(y0), active(fib(mark(x0)))))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(fib1(X1, X2)) → ACTIVE(fib1(mark(X1), mark(X2)))
ACTIVE(add(0, X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(mark(x0))), mark(y1)))

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule MARK(add(X1, X2)) → ACTIVE(add(mark(X1), mark(X2))) at position [0] we obtained the following new rules:

MARK(add(add(x0, x1), y1)) → ACTIVE(add(active(add(mark(x0), mark(x1))), mark(y1)))
MARK(add(fib(x0), y1)) → ACTIVE(add(active(fib(mark(x0))), mark(y1)))
MARK(add(y0, s(x0))) → ACTIVE(add(mark(y0), active(s(mark(x0)))))
MARK(add(y0, fib1(x0, x1))) → ACTIVE(add(mark(y0), active(fib1(mark(x0), mark(x1)))))
MARK(add(fib1(x0, x1), y1)) → ACTIVE(add(active(fib1(mark(x0), mark(x1))), mark(y1)))
MARK(add(s(x0), y1)) → ACTIVE(add(active(s(mark(x0))), mark(y1)))
MARK(add(y0, x1)) → ACTIVE(add(mark(y0), x1))
MARK(add(y0, sel(x0, x1))) → ACTIVE(add(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(add(cons(x0, x1), y1)) → ACTIVE(add(active(cons(mark(x0), x1)), mark(y1)))
MARK(add(y0, 0)) → ACTIVE(add(mark(y0), active(0)))
MARK(add(0, y1)) → ACTIVE(add(active(0), mark(y1)))
MARK(add(x0, y1)) → ACTIVE(add(x0, mark(y1)))
MARK(add(sel(x0, x1), y1)) → ACTIVE(add(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(add(y0, cons(x0, x1))) → ACTIVE(add(mark(y0), active(cons(mark(x0), x1))))
MARK(add(y0, fib(x0))) → ACTIVE(add(mark(y0), active(fib(mark(x0)))))
MARK(add(y0, add(x0, x1))) → ACTIVE(add(mark(y0), active(add(mark(x0), mark(x1)))))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

          ↳ QDP

            ↳ QDPOrderProof

              ↳ QDP

                ↳ Narrowing

                  ↳ QDP

                    ↳ Narrowing

                      ↳ QDP

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

ACTIVE(fib(N)) → MARK(sel(N, fib1(s(0), s(0))))
MARK(fib1(X1, X2)) → MARK(X1)
MARK(fib(X)) → ACTIVE(fib(mark(X)))
MARK(add(y0, s(x0))) → ACTIVE(add(mark(y0), active(s(mark(x0)))))
MARK(cons(X1, X2)) → MARK(X1)
MARK(add(y0, sel(x0, x1))) → ACTIVE(add(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(sel(y0, fib1(x0, x1))) → ACTIVE(sel(mark(y0), active(fib1(mark(x0), mark(x1)))))
ACTIVE(sel(s(N), cons(X, XS))) → MARK(sel(N, XS))
MARK(sel(fib(x0), y1)) → ACTIVE(sel(active(fib(mark(x0))), mark(y1)))
MARK(sel(y0, cons(x0, x1))) → ACTIVE(sel(mark(y0), active(cons(mark(x0), x1))))
MARK(add(sel(x0, x1), y1)) → ACTIVE(add(active(sel(mark(x0), mark(x1))), mark(y1)))
ACTIVE(sel(0, cons(X, XS))) → MARK(X)
MARK(sel(y0, s(x0))) → ACTIVE(sel(mark(y0), active(s(mark(x0)))))
MARK(add(y0, add(x0, x1))) → ACTIVE(add(mark(y0), active(add(mark(x0), mark(x1)))))
MARK(add(add(x0, x1), y1)) → ACTIVE(add(active(add(mark(x0), mark(x1))), mark(y1)))
MARK(sel(X1, X2)) → MARK(X1)
MARK(add(y0, x1)) → ACTIVE(add(mark(y0), x1))
MARK(sel(y0, sel(x0, x1))) → ACTIVE(sel(mark(y0), active(sel(mark(x0), mark(x1)))))
MARK(add(x0, y1)) → ACTIVE(add(x0, mark(y1)))
MARK(add(y0, cons(x0, x1))) → ACTIVE(add(mark(y0), active(cons(mark(x0), x1))))
MARK(fib(X)) → MARK(X)
MARK(s(X)) → MARK(X)
MARK(add(X1, X2)) → MARK(X2)
MARK(sel(fib1(x0, x1), y1)) → ACTIVE(sel(active(fib1(mark(x0), mark(x1))), mark(y1)))
MARK(fib1(X1, X2)) → MARK(X2)
ACTIVE(add(s(X), Y)) → MARK(s(add(X, Y)))
MARK(sel(cons(x0, x1), y1)) → ACTIVE(sel(active(cons(mark(x0), x1)), mark(y1)))
MARK(add(fib1(x0, x1), y1)) → ACTIVE(add(active(fib1(mark(x0), mark(x1))), mark(y1)))
MARK(sel(add(x0, x1), y1)) → ACTIVE(sel(active(add(mark(x0), mark(x1))), mark(y1)))
MARK(add(X1, X2)) → MARK(X1)
MARK(sel(y0, x1)) → ACTIVE(sel(mark(y0), x1))
MARK(add(cons(x0, x1), y1)) → ACTIVE(add(active(cons(mark(x0), x1)), mark(y1)))
MARK(sel(0, y1)) → ACTIVE(sel(active(0), mark(y1)))
MARK(sel(y0, 0)) → ACTIVE(sel(mark(y0), active(0)))
MARK(sel(y0, add(x0, x1))) → ACTIVE(sel(mark(y0), active(add(mark(x0), mark(x1)))))
MARK(sel(sel(x0, x1), y1)) → ACTIVE(sel(active(sel(mark(x0), mark(x1))), mark(y1)))
MARK(add(fib(x0), y1)) → ACTIVE(add(active(fib(mark(x0))), mark(y1)))
ACTIVE(fib1(X, Y)) → MARK(cons(X, fib1(Y, add(X, Y))))
MARK(add(y0, fib1(x0, x1))) → ACTIVE(add(mark(y0), active(fib1(mark(x0), mark(x1)))))
MARK(add(s(x0), y1)) → ACTIVE(add(active(s(mark(x0))), mark(y1)))
MARK(sel(y0, fib(x0))) → ACTIVE(sel(mark(y0), active(fib(mark(x0)))))
MARK(sel(x0, y1)) → ACTIVE(sel(x0, mark(y1)))
MARK(fib1(X1, X2)) → ACTIVE(fib1(mark(X1), mark(X2)))
MARK(add(y0, 0)) → ACTIVE(add(mark(y0), active(0)))
MARK(add(0, y1)) → ACTIVE(add(active(0), mark(y1)))
MARK(add(y0, fib(x0))) → ACTIVE(add(mark(y0), active(fib(mark(x0)))))
ACTIVE(add(0, X)) → MARK(X)
MARK(sel(X1, X2)) → MARK(X2)
MARK(sel(s(x0), y1)) → ACTIVE(sel(active(s(mark(x0))), mark(y1)))

The TRS R consists of the following rules:

active(fib(N)) → mark(sel(N, fib1(s(0), s(0))))
active(fib1(X, Y)) → mark(cons(X, fib1(Y, add(X, Y))))
active(add(0, X)) → mark(X)
active(add(s(X), Y)) → mark(s(add(X, Y)))
active(sel(0, cons(X, XS))) → mark(X)
active(sel(s(N), cons(X, XS))) → mark(sel(N, XS))
mark(fib(X)) → active(fib(mark(X)))
mark(sel(X1, X2)) → active(sel(mark(X1), mark(X2)))
mark(fib1(X1, X2)) → active(fib1(mark(X1), mark(X2)))
mark(s(X)) → active(s(mark(X)))
mark(0) → active(0)
mark(cons(X1, X2)) → active(cons(mark(X1), X2))
mark(add(X1, X2)) → active(add(mark(X1), mark(X2)))
fib(mark(X)) → fib(X)
fib(active(X)) → fib(X)
sel(mark(X1), X2) → sel(X1, X2)
sel(X1, mark(X2)) → sel(X1, X2)
sel(active(X1), X2) → sel(X1, X2)
sel(X1, active(X2)) → sel(X1, X2)
fib1(mark(X1), X2) → fib1(X1, X2)
fib1(X1, mark(X2)) → fib1(X1, X2)
fib1(active(X1), X2) → fib1(X1, X2)
fib1(X1, active(X2)) → fib1(X1, X2)
s(mark(X)) → s(X)
s(active(X)) → s(X)
cons(mark(X1), X2) → cons(X1, X2)
cons(X1, mark(X2)) → cons(X1, X2)
cons(active(X1), X2) → cons(X1, X2)
cons(X1, active(X2)) → cons(X1, X2)
add(mark(X1), X2) → add(X1, X2)
add(X1, mark(X2)) → add(X1, X2)
add(active(X1), X2) → add(X1, X2)
add(X1, active(X2)) → add(X1, X2)

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