Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Secret_06

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 QTRS Reverse (⇔, 0 ms)

↳2 QTRS

↳3 FlatCCProof (⇔, 0 ms)

↳4 QTRS

↳5 RootLabelingProof (⇔, 0 ms)

↳6 QTRS

↳7 DependencyPairsProof (⇔, 25 ms)

↳8 QDP

↳9 QDPOrderProof (⇔, 69 ms)

↳10 QDP

↳11 DependencyGraphProof (⇔, 0 ms)

↳12 QDP

↳13 QDPOrderProof (⇔, 30 ms)

↳14 QDP

↳15 QDPOrderProof (⇔, 21 ms)

↳16 QDP

↳17 PisEmptyProof (⇔, 0 ms)

↳18 YES

(0) Obligation:

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

a(a(a(b(x)))) → b(a(a(a(x))))
b(a(b(a(x)))) → a(b(b(a(x))))

Q is empty.

(1) QTRS Reverse (EQUIVALENT transformation)

We applied the QTRS Reverse Processor [REVERSE].

(2) Obligation:

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

b(a(a(a(x)))) → a(a(a(b(x))))
a(b(a(b(x)))) → a(b(b(a(x))))

Q is empty.

(3) FlatCCProof (EQUIVALENT transformation)

We used flat context closure [ROOTLAB] As Q is empty the flat context closure was sound AND complete.

(4) Obligation:

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

a(b(a(b(x)))) → a(b(b(a(x))))
b(b(a(a(a(x))))) → b(a(a(a(b(x)))))
a(b(a(a(a(x))))) → a(a(a(a(b(x)))))

Q is empty.

(5) RootLabelingProof (EQUIVALENT transformation)

We used plain root labeling [ROOTLAB] with the following heuristic: LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.

(6) Obligation:

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

a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))

Q is empty.

(7) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(8) Obligation:

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

A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{a_1}(a_{a_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{a_1}(a_{b_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → A_{B_1}(x)
B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{a_1}(x))
B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(x))
B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{a_1}(x))
A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(x))
A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)

The TRS R consists of the following rules:

a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))

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

(9) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → B_{B_1}(b_{a_1}(a_{a_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → B_{B_1}(b_{a_1}(a_{b_1}(x)))
A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → A_{B_1}(x)
B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)
A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → B_{B_1}(x)

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A_{B_1}(x₁)) = x₁
POL(B_{B_1}(x₁)) = x₁
POL(a_{a_1}(x₁)) = x₁
POL(a_{b_1}(x₁)) = 1 + x₁
POL(b_{a_1}(x₁)) = x₁
POL(b_{b_1}(x₁)) = 1 + x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))

(10) Obligation:

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

A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{a_1}(x))
B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(x))
A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{a_1}(x))
A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(x))

The TRS R consists of the following rules:

a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))

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

(11) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(12) Obligation:

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

A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{a_1}(x))
A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(x))

The TRS R consists of the following rules:

a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))

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

(13) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → A_{B_1}(b_{a_1}(x))
A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → A_{B_1}(b_{b_1}(x))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A_{B_1}(x₁)) = x₁
POL(a_{a_1}(x₁)) = 1 + x₁
POL(a_{b_1}(x₁)) = 1 + x₁
POL(b_{a_1}(x₁)) = x₁
POL(b_{b_1}(x₁)) = x₁

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))

(14) Obligation:

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

A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))

The TRS R consists of the following rules:

a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))

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

(15) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))

The remaining pairs can at least be oriented weakly.
Used ordering: Polynomial interpretation [POLO]:

POL(A_{B_1}(x₁)) = x₁
POL(a_{a_1}(x₁)) = 0
POL(a_{b_1}(x₁)) = 1
POL(b_{a_1}(x₁)) = x₁
POL(b_{b_1}(x₁)) = 0

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:

b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))

(16) Obligation:

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

a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x))))
a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x)))) → a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x)))))
a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x))))) → a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x)))))

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

(17) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(0) Obligation:

(1) QTRS Reverse (EQUIVALENT transformation)

(2) Obligation:

(3) FlatCCProof (EQUIVALENT transformation)

(4) Obligation:

(5) RootLabelingProof (EQUIVALENT transformation)

(6) Obligation:

(7) DependencyPairsProof (EQUIVALENT transformation)

(8) Obligation:

(9) QDPOrderProof (EQUIVALENT transformation)

(10) Obligation:

(11) DependencyGraphProof (EQUIVALENT transformation)

(12) Obligation:

(13) QDPOrderProof (EQUIVALENT transformation)

(14) Obligation:

(15) QDPOrderProof (EQUIVALENT transformation)

(16) Obligation:

(17) PisEmptyProof (EQUIVALENT transformation)

(18) YES