YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z002.srs

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

0 QTRS
↳1 QTRS Reverse (⇔, 0 ms)
↳2 QTRS
↳3 RFCMatchBoundsTRSProof (⇔, 0 ms)
↳4 YES

(0) Obligation:

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

b(c(a(x))) → a(b(a(b(x))))
b(x) → c(c(x))
a(a(x)) → a(c(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:

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

Q is empty.

(3) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 2. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 18, 19, 21, 24, 25, 26, 27, 28, 29, 30

Node 2 is start node and node 4 is final node.

Those nodes are connected through the following edges:

  • 2 to 5 labelled b_1(0)
  • 2 to 8 labelled c_1(0)
  • 2 to 9 labelled a_1(0)
  • 2 to 12 labelled c_1(1)
  • 4 to 4 labelled #_1(0)
  • 5 to 6 labelled a_1(0)
  • 6 to 7 labelled b_1(0)
  • 6 to 16 labelled c_1(1)
  • 7 to 4 labelled a_1(0)
  • 7 to 19 labelled b_1(1)
  • 7 to 25 labelled a_1(1)
  • 7 to 28 labelled c_1(2)
  • 8 to 4 labelled c_1(0)
  • 9 to 10 labelled b_1(0)
  • 9 to 18 labelled c_1(1)
  • 10 to 11 labelled c_1(0)
  • 11 to 4 labelled a_1(0)
  • 11 to 19 labelled b_1(1)
  • 11 to 25 labelled a_1(1)
  • 11 to 28 labelled c_1(2)
  • 12 to 5 labelled c_1(1)
  • 16 to 7 labelled c_1(1)
  • 18 to 10 labelled c_1(1)
  • 19 to 21 labelled a_1(1)
  • 21 to 24 labelled b_1(1)
  • 21 to 29 labelled c_1(2)
  • 24 to 4 labelled a_1(1)
  • 24 to 19 labelled b_1(1)
  • 24 to 25 labelled a_1(1)
  • 24 to 28 labelled c_1(2)
  • 25 to 26 labelled b_1(1)
  • 25 to 30 labelled c_1(2)
  • 26 to 27 labelled c_1(1)
  • 27 to 4 labelled a_1(1)
  • 27 to 19 labelled b_1(1)
  • 27 to 25 labelled a_1(1)
  • 27 to 28 labelled c_1(2)
  • 28 to 19 labelled c_1(2)
  • 29 to 24 labelled c_1(2)
  • 30 to 26 labelled c_1(2)

(4) YES