YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/tpdb/TPDB-d9b80194f163/SRS_Standard/Zantema_04/z124.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:

q0(a(x1)) → x(q1(x1))
q1(a(x1)) → a(q1(x1))
q1(y(x1)) → y(q1(x1))
a(q1(b(x1))) → q2(a(y(x1)))
a(q2(a(x1))) → q2(a(a(x1)))
a(q2(y(x1))) → q2(a(y(x1)))
y(q1(b(x1))) → q2(y(y(x1)))
y(q2(a(x1))) → q2(y(a(x1)))
y(q2(y(x1))) → q2(y(y(x1)))
q2(x(x1)) → x(q0(x1))
q0(y(x1)) → y(q3(x1))
q3(y(x1)) → y(q3(x1))
q3(bl(x1)) → bl(q4(x1))

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(q0(x1)) → q1(x(x1))
a(q1(x1)) → q1(a(x1))
y(q1(x1)) → q1(y(x1))
b(q1(a(x1))) → y(a(q2(x1)))
a(q2(a(x1))) → a(a(q2(x1)))
y(q2(a(x1))) → y(a(q2(x1)))
b(q1(y(x1))) → y(y(q2(x1)))
a(q2(y(x1))) → a(y(q2(x1)))
y(q2(y(x1))) → y(y(q2(x1)))
x(q2(x1)) → q0(x(x1))
y(q0(x1)) → q3(y(x1))
y(q3(x1)) → q3(y(x1))
bl(q3(x1)) → q4(bl(x1))

Q is empty.

(3) 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(q0(x1)) → q1(x(x1))
a(q1(x1)) → q1(a(x1))
y(q1(x1)) → q1(y(x1))
b(q1(a(x1))) → y(a(q2(x1)))
a(q2(a(x1))) → a(a(q2(x1)))
y(q2(a(x1))) → y(a(q2(x1)))
b(q1(y(x1))) → y(y(q2(x1)))
a(q2(y(x1))) → a(y(q2(x1)))
y(q2(y(x1))) → y(y(q2(x1)))
x(q2(x1)) → q0(x(x1))
y(q0(x1)) → q3(y(x1))
y(q3(x1)) → q3(y(x1))
bl(q3(x1)) → q4(bl(x1))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 2, 5, 6, 7, 9, 10, 11, 17, 18, 19, 21, 22, 26

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

Those nodes are connected through the following edges:

  • 1 to 5 labelled q1_1(0), q0_1(0), q3_1(0)
  • 1 to 6 labelled y_1(0), a_1(0)
  • 1 to 9 labelled y_1(0), a_1(0)
  • 1 to 11 labelled q4_1(0)
  • 2 to 2 labelled #_1(0)
  • 5 to 2 labelled x_1(0), a_1(0), y_1(0)
  • 5 to 17 labelled q0_1(1), q1_1(1), q3_1(1)
  • 5 to 18 labelled a_1(1), y_1(1)
  • 5 to 21 labelled a_1(1), y_1(1)
  • 6 to 7 labelled a_1(0)
  • 6 to 18 labelled a_1(1)
  • 6 to 21 labelled a_1(1)
  • 7 to 2 labelled q2_1(0)
  • 9 to 10 labelled y_1(0)
  • 9 to 18 labelled y_1(1)
  • 9 to 21 labelled y_1(1)
  • 10 to 2 labelled q2_1(0)
  • 11 to 2 labelled bl_1(0)
  • 11 to 26 labelled q4_1(1)
  • 17 to 2 labelled x_1(1), a_1(1), y_1(1)
  • 17 to 17 labelled q0_1(1), q1_1(1), q3_1(1)
  • 17 to 18 labelled a_1(1), y_1(1)
  • 17 to 21 labelled a_1(1), y_1(1)
  • 18 to 19 labelled a_1(1)
  • 18 to 18 labelled a_1(1)
  • 18 to 21 labelled a_1(1)
  • 19 to 2 labelled q2_1(1)
  • 21 to 22 labelled y_1(1)
  • 21 to 18 labelled y_1(1)
  • 21 to 21 labelled y_1(1)
  • 22 to 2 labelled q2_1(1)
  • 26 to 2 labelled bl_1(1)
  • 26 to 26 labelled q4_1(1)

(4) YES