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 Strip Symbols Proof (⇒, 0 ms)

↳4 QTRS

↳5 RFCMatchBoundsTRSProof (⇔, 0 ms)

↳6 YES

(0) Obligation:

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

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

Q is empty.

(3) Strip Symbols Proof (SOUND transformation)

We were given the following TRS:

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

By stripping symbols from the only rule of the system, we obtained the following TRS [ENDRULLIS]:

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

(4) Obligation:

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

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

Q is empty.

(5) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

7, 8, 9, 11, 13, 15, 17, 19, 22, 24, 27, 29, 31, 33, 35, 37, 39, 40, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106

Node 7 is start node and node 8 is final node.

Those nodes are connected through the following edges:

7 to 9 labelled b_1(0)
8 to 8 labelled #_1(0)
9 to 11 labelled b_1(0)
11 to 13 labelled b_1(0)
13 to 15 labelled b_1(0)
15 to 17 labelled b_1(0)
17 to 19 labelled a_1(0)
19 to 22 labelled b_1(0)
22 to 24 labelled b_1(0)
24 to 27 labelled a_1(0)
27 to 29 labelled b_1(0)
29 to 31 labelled b_1(0)
31 to 33 labelled a_1(0)
31 to 91 labelled b_1(1)
33 to 35 labelled b_1(0)
35 to 37 labelled b_1(0)
37 to 39 labelled b_1(0)
39 to 40 labelled b_1(0)
40 to 8 labelled b_1(0)
91 to 92 labelled b_1(1)
92 to 93 labelled b_1(1)
93 to 94 labelled b_1(1)
94 to 95 labelled b_1(1)
95 to 96 labelled a_1(1)
96 to 97 labelled b_1(1)
97 to 98 labelled b_1(1)
98 to 99 labelled a_1(1)
99 to 100 labelled b_1(1)
100 to 101 labelled b_1(1)
101 to 102 labelled a_1(1)
101 to 91 labelled b_1(1)
102 to 103 labelled b_1(1)
103 to 104 labelled b_1(1)
104 to 105 labelled b_1(1)
105 to 106 labelled b_1(1)
106 to 8 labelled b_1(1)

(0) Obligation:

(1) QTRS Reverse (EQUIVALENT transformation)

(2) Obligation:

(3) Strip Symbols Proof (SOUND transformation)

(4) Obligation:

(5) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

(6) YES