Termination w.r.t. Q proof of /tmp/tmp32_vPR/Ex9_Luc06

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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

active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
mark(f(X1, X2, X3)) → active(f(X1, mark(X2), X3))
mark(a) → active(a)
mark(b) → active(b)
f(mark(X1), X2, X3) → f(X1, X2, X3)
f(X1, mark(X2), X3) → f(X1, X2, X3)
f(X1, X2, mark(X3)) → f(X1, X2, X3)
f(active(X1), X2, X3) → f(X1, X2, X3)
f(X1, active(X2), X3) → f(X1, X2, X3)
f(X1, X2, active(X3)) → f(X1, X2, X3)

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
mark(f(X1, X2, X3)) → active(f(X1, mark(X2), X3))
mark(a) → active(a)
mark(b) → active(b)
f(mark(X1), X2, X3) → f(X1, X2, X3)
f(X1, mark(X2), X3) → f(X1, X2, X3)
f(X1, X2, mark(X3)) → f(X1, X2, X3)
f(active(X1), X2, X3) → f(X1, X2, X3)
f(X1, active(X2), X3) → f(X1, X2, X3)
f(X1, X2, active(X3)) → f(X1, X2, X3)

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:

ACTIVE(f(a, X, X)) → F(X, b, b)
MARK(f(X1, X2, X3)) → MARK(X2)
ACTIVE(b) → MARK(a)
F(X1, mark(X2), X3) → F(X1, X2, X3)
MARK(b) → ACTIVE(b)
F(X1, X2, mark(X3)) → F(X1, X2, X3)
F(active(X1), X2, X3) → F(X1, X2, X3)
ACTIVE(f(a, X, X)) → MARK(f(X, b, b))
MARK(a) → ACTIVE(a)
MARK(f(X1, X2, X3)) → F(X1, mark(X2), X3)
F(X1, active(X2), X3) → F(X1, X2, X3)
F(X1, X2, active(X3)) → F(X1, X2, X3)
F(mark(X1), X2, X3) → F(X1, X2, X3)
MARK(f(X1, X2, X3)) → ACTIVE(f(X1, mark(X2), X3))

The TRS R consists of the following rules:

active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
mark(f(X1, X2, X3)) → active(f(X1, mark(X2), X3))
mark(a) → active(a)
mark(b) → active(b)
f(mark(X1), X2, X3) → f(X1, X2, X3)
f(X1, mark(X2), X3) → f(X1, X2, X3)
f(X1, X2, mark(X3)) → f(X1, X2, X3)
f(active(X1), X2, X3) → f(X1, X2, X3)
f(X1, active(X2), X3) → f(X1, X2, X3)
f(X1, X2, active(X3)) → f(X1, X2, X3)

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:

ACTIVE(f(a, X, X)) → F(X, b, b)
MARK(f(X1, X2, X3)) → MARK(X2)
ACTIVE(b) → MARK(a)
F(X1, mark(X2), X3) → F(X1, X2, X3)
MARK(b) → ACTIVE(b)
F(X1, X2, mark(X3)) → F(X1, X2, X3)
F(active(X1), X2, X3) → F(X1, X2, X3)
ACTIVE(f(a, X, X)) → MARK(f(X, b, b))
MARK(a) → ACTIVE(a)
MARK(f(X1, X2, X3)) → F(X1, mark(X2), X3)
F(X1, active(X2), X3) → F(X1, X2, X3)
F(X1, X2, active(X3)) → F(X1, X2, X3)
F(mark(X1), X2, X3) → F(X1, X2, X3)
MARK(f(X1, X2, X3)) → ACTIVE(f(X1, mark(X2), X3))

The TRS R consists of the following rules:

active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
mark(f(X1, X2, X3)) → active(f(X1, mark(X2), X3))
mark(a) → active(a)
mark(b) → active(b)
f(mark(X1), X2, X3) → f(X1, X2, X3)
f(X1, mark(X2), X3) → f(X1, X2, X3)
f(X1, X2, mark(X3)) → f(X1, X2, X3)
f(active(X1), X2, X3) → f(X1, X2, X3)
f(X1, active(X2), X3) → f(X1, X2, X3)
f(X1, X2, active(X3)) → f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs with 5 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

            ↳ UsableRulesProof

          ↳ QDP

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

F(X1, active(X2), X3) → F(X1, X2, X3)
F(X1, X2, active(X3)) → F(X1, X2, X3)
F(mark(X1), X2, X3) → F(X1, X2, X3)
F(X1, mark(X2), X3) → F(X1, X2, X3)
F(X1, X2, mark(X3)) → F(X1, X2, X3)
F(active(X1), X2, X3) → F(X1, X2, X3)

The TRS R consists of the following rules:

active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
mark(f(X1, X2, X3)) → active(f(X1, mark(X2), X3))
mark(a) → active(a)
mark(b) → active(b)
f(mark(X1), X2, X3) → f(X1, X2, X3)
f(X1, mark(X2), X3) → f(X1, X2, X3)
f(X1, X2, mark(X3)) → f(X1, X2, X3)
f(active(X1), X2, X3) → f(X1, X2, X3)
f(X1, active(X2), X3) → f(X1, X2, X3)
f(X1, X2, active(X3)) → f(X1, X2, X3)

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

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

F(X1, active(X2), X3) → F(X1, X2, X3)
F(mark(X1), X2, X3) → F(X1, X2, X3)
F(X1, X2, active(X3)) → F(X1, X2, X3)
F(X1, mark(X2), X3) → F(X1, X2, X3)
F(X1, X2, mark(X3)) → F(X1, X2, X3)
F(active(X1), X2, X3) → F(X1, X2, X3)

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:

F(X1, active(X2), X3) → F(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
F(X1, X2, active(X3)) → F(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
F(mark(X1), X2, X3) → F(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
F(X1, mark(X2), X3) → F(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
F(X1, X2, mark(X3)) → F(X1, X2, X3)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3
F(active(X1), X2, X3) → F(X1, X2, X3)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ RuleRemovalProof

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

ACTIVE(f(a, X, X)) → MARK(f(X, b, b))
MARK(f(X1, X2, X3)) → MARK(X2)
MARK(f(X1, X2, X3)) → ACTIVE(f(X1, mark(X2), X3))

The TRS R consists of the following rules:

active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
mark(f(X1, X2, X3)) → active(f(X1, mark(X2), X3))
mark(a) → active(a)
mark(b) → active(b)
f(mark(X1), X2, X3) → f(X1, X2, X3)
f(X1, mark(X2), X3) → f(X1, X2, X3)
f(X1, X2, mark(X3)) → f(X1, X2, X3)
f(active(X1), X2, X3) → f(X1, X2, X3)
f(X1, active(X2), X3) → f(X1, X2, X3)
f(X1, X2, active(X3)) → f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

MARK(f(X1, X2, X3)) → MARK(X2)

Used ordering: POLO with Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = x₁
POL(a) = 0
POL(active(x₁)) = x₁
POL(b) = 0
POL(f(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(mark(x₁)) = x₁

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ RuleRemovalProof

              ↳ QDP

                ↳ QDPOrderProof

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

ACTIVE(f(a, X, X)) → MARK(f(X, b, b))
MARK(f(X1, X2, X3)) → ACTIVE(f(X1, mark(X2), X3))

The TRS R consists of the following rules:

active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
mark(f(X1, X2, X3)) → active(f(X1, mark(X2), X3))
mark(a) → active(a)
mark(b) → active(b)
f(mark(X1), X2, X3) → f(X1, X2, X3)
f(X1, mark(X2), X3) → f(X1, X2, X3)
f(X1, X2, mark(X3)) → f(X1, X2, X3)
f(active(X1), X2, X3) → f(X1, X2, X3)
f(X1, active(X2), X3) → f(X1, X2, X3)
f(X1, X2, active(X3)) → f(X1, X2, X3)

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(f(X1, X2, X3)) → ACTIVE(f(X1, mark(X2), X3))

The remaining pairs can at least be oriented weakly.

ACTIVE(f(a, X, X)) → MARK(f(X, b, b))

Used ordering: Polynomial interpretation [25]:

POL(ACTIVE(x₁)) = x₁
POL(MARK(x₁)) = 1 + x₁
POL(a) = 1
POL(active(x₁)) = x₁
POL(b) = 0
POL(f(x₁, x₂, x₃)) = 1 + x₁ + x₃
POL(mark(x₁)) = x₁

The following usable rules [17] were oriented:

f(X1, X2, active(X3)) → f(X1, X2, X3)
f(mark(X1), X2, X3) → f(X1, X2, X3)
f(active(X1), X2, X3) → f(X1, X2, X3)
f(X1, active(X2), X3) → f(X1, X2, X3)
f(X1, X2, mark(X3)) → f(X1, X2, X3)
f(X1, mark(X2), X3) → f(X1, X2, X3)

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ AND

          ↳ QDP

          ↳ QDP

            ↳ RuleRemovalProof

              ↳ QDP

                ↳ QDPOrderProof

                  ↳ QDP

                    ↳ DependencyGraphProof

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

ACTIVE(f(a, X, X)) → MARK(f(X, b, b))

The TRS R consists of the following rules:

active(f(a, X, X)) → mark(f(X, b, b))
active(b) → mark(a)
mark(f(X1, X2, X3)) → active(f(X1, mark(X2), X3))
mark(a) → active(a)
mark(b) → active(b)
f(mark(X1), X2, X3) → f(X1, X2, X3)
f(X1, mark(X2), X3) → f(X1, X2, X3)
f(X1, X2, mark(X3)) → f(X1, X2, X3)
f(active(X1), X2, X3) → f(X1, X2, X3)
f(X1, active(X2), X3) → f(X1, X2, X3)
f(X1, X2, active(X3)) → f(X1, X2, X3)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.