Termination w.r.t. Q proof of /tmp/tmpyxYmy8/matchbox1.srs

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:

r(e(x1)) → w(r(x1))
i(t(x1)) → e(r(x1))
e(w(x1)) → r(i(x1))
t(e(x1)) → r(e(x1))
w(r(x1)) → i(t(x1))
e(r(x1)) → e(w(x1))
r(i(t(e(r(x1))))) → e(w(r(i(t(e(x1))))))

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

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

r(e(x1)) → w(r(x1))
i(t(x1)) → e(r(x1))
e(w(x1)) → r(i(x1))
t(e(x1)) → r(e(x1))
w(r(x1)) → i(t(x1))
e(r(x1)) → e(w(x1))
r(i(t(e(r(x1))))) → e(w(r(i(t(e(x1))))))

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:

E(w(x1)) → R(i(x1))
E(r(x1)) → W(x1)
R(i(t(e(r(x1))))) → W(r(i(t(e(x1)))))
R(i(t(e(r(x1))))) → I(t(e(x1)))
R(e(x1)) → W(r(x1))
W(r(x1)) → T(x1)
E(w(x1)) → I(x1)
R(i(t(e(r(x1))))) → E(w(r(i(t(e(x1))))))
I(t(x1)) → R(x1)
R(i(t(e(r(x1))))) → T(e(x1))
W(r(x1)) → I(t(x1))
I(t(x1)) → E(r(x1))
R(e(x1)) → R(x1)
R(i(t(e(r(x1))))) → R(i(t(e(x1))))
R(i(t(e(r(x1))))) → E(x1)
E(r(x1)) → E(w(x1))
T(e(x1)) → R(e(x1))

The TRS R consists of the following rules:

r(e(x1)) → w(r(x1))
i(t(x1)) → e(r(x1))
e(w(x1)) → r(i(x1))
t(e(x1)) → r(e(x1))
w(r(x1)) → i(t(x1))
e(r(x1)) → e(w(x1))
r(i(t(e(r(x1))))) → e(w(r(i(t(e(x1))))))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

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

E(w(x1)) → R(i(x1))
E(r(x1)) → W(x1)
R(i(t(e(r(x1))))) → W(r(i(t(e(x1)))))
R(i(t(e(r(x1))))) → I(t(e(x1)))
R(e(x1)) → W(r(x1))
W(r(x1)) → T(x1)
E(w(x1)) → I(x1)
R(i(t(e(r(x1))))) → E(w(r(i(t(e(x1))))))
I(t(x1)) → R(x1)
R(i(t(e(r(x1))))) → T(e(x1))
W(r(x1)) → I(t(x1))
I(t(x1)) → E(r(x1))
R(e(x1)) → R(x1)
R(i(t(e(r(x1))))) → R(i(t(e(x1))))
R(i(t(e(r(x1))))) → E(x1)
E(r(x1)) → E(w(x1))
T(e(x1)) → R(e(x1))

The TRS R consists of the following rules:

r(e(x1)) → w(r(x1))
i(t(x1)) → e(r(x1))
e(w(x1)) → r(i(x1))
t(e(x1)) → r(e(x1))
w(r(x1)) → i(t(x1))
e(r(x1)) → e(w(x1))
r(i(t(e(r(x1))))) → e(w(r(i(t(e(x1))))))

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.

E(r(x1)) → W(x1)
R(i(t(e(r(x1))))) → W(r(i(t(e(x1)))))
R(i(t(e(r(x1))))) → I(t(e(x1)))
W(r(x1)) → T(x1)
E(w(x1)) → I(x1)
I(t(x1)) → R(x1)
R(i(t(e(r(x1))))) → T(e(x1))
R(e(x1)) → R(x1)
R(i(t(e(r(x1))))) → R(i(t(e(x1))))
R(i(t(e(r(x1))))) → E(x1)
E(r(x1)) → E(w(x1))

The remaining pairs can at least be oriented weakly.

E(w(x1)) → R(i(x1))
R(e(x1)) → W(r(x1))
R(i(t(e(r(x1))))) → E(w(r(i(t(e(x1))))))
W(r(x1)) → I(t(x1))
I(t(x1)) → E(r(x1))
T(e(x1)) → R(e(x1))

Used ordering: Polynomial interpretation [25,35]:

POL(E(x₁)) = 1 + x_1
POL(i(x₁)) = x_1
POL(T(x₁)) = 2 + x_1
POL(w(x₁)) = 1 + x_1
POL(t(x₁)) = 3 + x_1
POL(e(x₁)) = 1 + x_1
POL(r(x₁)) = 2 + x_1
POL(R(x₁)) = 2 + x_1
POL(I(x₁)) = x_1
POL(W(x₁)) = 1 + x_1

The value of delta used in the strict ordering is 1.
The following usable rules [17] were oriented:

t(e(x1)) → r(e(x1))
e(r(x1)) → e(w(x1))
e(w(x1)) → r(i(x1))
r(i(t(e(r(x1))))) → e(w(r(i(t(e(x1))))))
i(t(x1)) → e(r(x1))
r(e(x1)) → w(r(x1))
w(r(x1)) → i(t(x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

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

I(t(x1)) → E(r(x1))
E(w(x1)) → R(i(x1))
R(e(x1)) → W(r(x1))
R(i(t(e(r(x1))))) → E(w(r(i(t(e(x1))))))
T(e(x1)) → R(e(x1))
W(r(x1)) → I(t(x1))

The TRS R consists of the following rules:

r(e(x1)) → w(r(x1))
i(t(x1)) → e(r(x1))
e(w(x1)) → r(i(x1))
t(e(x1)) → r(e(x1))
w(r(x1)) → i(t(x1))
e(r(x1)) → e(w(x1))
r(i(t(e(r(x1))))) → e(w(r(i(t(e(x1))))))

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

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ QDPOrderProof

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

E(w(x1)) → R(i(x1))
I(t(x1)) → E(r(x1))
R(e(x1)) → W(r(x1))
R(i(t(e(r(x1))))) → E(w(r(i(t(e(x1))))))
W(r(x1)) → I(t(x1))

The TRS R consists of the following rules:

r(e(x1)) → w(r(x1))
i(t(x1)) → e(r(x1))
e(w(x1)) → r(i(x1))
t(e(x1)) → r(e(x1))
w(r(x1)) → i(t(x1))
e(r(x1)) → e(w(x1))
r(i(t(e(r(x1))))) → e(w(r(i(t(e(x1))))))

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.

I(t(x1)) → E(r(x1))
R(i(t(e(r(x1))))) → E(w(r(i(t(e(x1))))))

The remaining pairs can at least be oriented weakly.

E(w(x1)) → R(i(x1))
R(e(x1)) → W(r(x1))
W(r(x1)) → I(t(x1))

Used ordering: Polynomial interpretation [25,35]:

POL(E(x₁)) = (1/4)x_1
POL(i(x₁)) = (1/4)x_1
POL(w(x₁)) = x_1
POL(t(x₁)) = 1
POL(e(x₁)) = 1/4
POL(r(x₁)) = 1/4
POL(R(x₁)) = x_1
POL(I(x₁)) = 1/4
POL(W(x₁)) = 1/4

The value of delta used in the strict ordering is 3/16.
The following usable rules [17] were oriented:

t(e(x1)) → r(e(x1))
e(r(x1)) → e(w(x1))
e(w(x1)) → r(i(x1))
r(i(t(e(r(x1))))) → e(w(r(i(t(e(x1))))))
i(t(x1)) → e(r(x1))
r(e(x1)) → w(r(x1))
w(r(x1)) → i(t(x1))

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ QDPOrderProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ DependencyGraphProof

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

E(w(x1)) → R(i(x1))
R(e(x1)) → W(r(x1))
W(r(x1)) → I(t(x1))

The TRS R consists of the following rules:

r(e(x1)) → w(r(x1))
i(t(x1)) → e(r(x1))
e(w(x1)) → r(i(x1))
t(e(x1)) → r(e(x1))
w(r(x1)) → i(t(x1))
e(r(x1)) → e(w(x1))
r(i(t(e(r(x1))))) → e(w(r(i(t(e(x1))))))

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 3 less nodes.