YES Termination w.r.t. Q proof of /home/cern_httpd/provide/research/cycsrs/examples/collection/a-b-split.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:

Aa(Left(x)) → Left(a(x))
Ab(Left(x)) → Left(b(x))
Wait(Left(x)) → Begin(x)
a(x) → 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:

Left(Aa(x)) → a(Left(x))
Left(Ab(x)) → b(Left(x))
Left(Wait(x)) → Begin(x)
a(x) → b(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:

Left(Aa(x)) → a(Left(x))
Left(Ab(x)) → b(Left(x))
Left(Wait(x)) → Begin(x)
a(x) → b(x)

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

2, 4, 6, 7

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

Those nodes are connected through the following edges:

  • 2 to 6 labelled a_1(0), b_1(0), b_1(1)
  • 2 to 4 labelled Begin_1(0), b_1(0)
  • 4 to 4 labelled #_1(0)
  • 6 to 4 labelled Left_1(0), Begin_1(1)
  • 6 to 7 labelled a_1(1), b_1(1), b_1(2)
  • 7 to 4 labelled Left_1(1), Begin_1(1)
  • 7 to 7 labelled a_1(1), b_1(1), b_1(2)

(4) YES