YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z014.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(x) → c(a(c(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)))
b(x) → c(a(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)))
b(x) → c(a(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:

3, 4, 5, 7, 9, 10, 13, 14, 18, 20, 23, 24, 27, 28, 29, 30, 55, 56, 57, 58, 59, 60, 75, 76, 77, 78, 79, 80

Node 3 is start node and node 4 is final node.

Those nodes are connected through the following edges:

  • 3 to 5 labelled a_1(0)
  • 3 to 9 labelled c_1(0)
  • 3 to 13 labelled a_1(0)
  • 3 to 23 labelled a_1(1)
  • 4 to 4 labelled #_1(0)
  • 5 to 7 labelled a_1(0)
  • 5 to 57 labelled a_1(1)
  • 7 to 4 labelled b_1(0)
  • 7 to 18 labelled c_1(1)
  • 7 to 27 labelled a_1(1)
  • 7 to 75 labelled a_1(2)
  • 9 to 10 labelled a_1(0)
  • 10 to 4 labelled c_1(0)
  • 13 to 14 labelled c_1(0)
  • 14 to 4 labelled a_1(0)
  • 14 to 29 labelled a_1(1)
  • 18 to 20 labelled a_1(1)
  • 20 to 4 labelled c_1(1)
  • 23 to 24 labelled c_1(1)
  • 24 to 7 labelled a_1(1)
  • 24 to 59 labelled a_1(2)
  • 24 to 57 labelled a_1(1)
  • 27 to 28 labelled a_1(1)
  • 27 to 59 labelled a_1(2)
  • 28 to 4 labelled b_1(1)
  • 28 to 55 labelled c_1(2)
  • 28 to 27 labelled a_1(1)
  • 28 to 75 labelled a_1(2)
  • 29 to 30 labelled c_1(1)
  • 30 to 4 labelled a_1(1)
  • 30 to 29 labelled a_1(1)
  • 55 to 56 labelled a_1(2)
  • 56 to 4 labelled c_1(2)
  • 57 to 58 labelled c_1(1)
  • 58 to 27 labelled a_1(1)
  • 58 to 75 labelled a_1(1), a_1(2)
  • 59 to 60 labelled c_1(2)
  • 60 to 27 labelled a_1(2)
  • 60 to 75 labelled a_1(2)
  • 60 to 77 labelled a_1(3)
  • 75 to 76 labelled c_1(2)
  • 76 to 28 labelled a_1(2)
  • 76 to 59 labelled a_1(2)
  • 76 to 79 labelled a_1(3)
  • 77 to 78 labelled c_1(3)
  • 78 to 59 labelled a_1(3)
  • 79 to 80 labelled c_1(3)
  • 80 to 75 labelled a_1(3)

(4) YES