YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z017.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:

a(b(x)) → b(a(a(x)))
b(c(x)) → c(b(x))
a(a(x)) → a(c(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(x)) → a(a(b(x)))
c(b(x)) → b(c(x))
a(a(x)) → a(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 3. This implies Q-termination of R.
The following rules were used to construct the certificate:

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

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 103, 104, 105, 106

Node 1 is start node and node 2 is final node.

Those nodes are connected through the following edges:

  • 1 to 3 labelled a_1(0)
  • 1 to 5 labelled b_1(0)
  • 1 to 6 labelled a_1(0)
  • 1 to 8 labelled a_1(1)
  • 2 to 2 labelled #_1(0)
  • 3 to 4 labelled a_1(0)
  • 3 to 15 labelled a_1(1)
  • 4 to 2 labelled b_1(0)
  • 4 to 10 labelled a_1(1)
  • 4 to 19 labelled a_1(2)
  • 5 to 2 labelled c_1(0)
  • 5 to 12 labelled b_1(1)
  • 6 to 7 labelled c_1(0)
  • 7 to 2 labelled a_1(0)
  • 7 to 13 labelled a_1(1)
  • 8 to 9 labelled c_1(1)
  • 9 to 4 labelled a_1(1)
  • 9 to 17 labelled a_1(2)
  • 9 to 15 labelled a_1(1)
  • 10 to 11 labelled a_1(1)
  • 10 to 17 labelled a_1(2)
  • 11 to 2 labelled b_1(1)
  • 11 to 10 labelled a_1(1)
  • 11 to 19 labelled a_1(2)
  • 12 to 2 labelled c_1(1)
  • 12 to 12 labelled b_1(1)
  • 13 to 14 labelled c_1(1)
  • 14 to 2 labelled a_1(1)
  • 14 to 13 labelled a_1(1)
  • 15 to 16 labelled c_1(1)
  • 16 to 10 labelled a_1(1)
  • 16 to 19 labelled a_1(1), a_1(2)
  • 17 to 18 labelled c_1(2)
  • 18 to 10 labelled a_1(2)
  • 18 to 19 labelled a_1(2)
  • 18 to 103 labelled a_1(3)
  • 19 to 20 labelled c_1(2)
  • 20 to 11 labelled a_1(2)
  • 20 to 17 labelled a_1(2)
  • 20 to 105 labelled a_1(3)
  • 103 to 104 labelled c_1(3)
  • 104 to 17 labelled a_1(3)
  • 105 to 106 labelled c_1(3)
  • 106 to 19 labelled a_1(3)

(4) YES