YES Termination w.r.t. Q proof of CiME_04_dpqs.ari

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

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

(0) Obligation:

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

f(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(0)) → g(d(1))
g(c(1)) → g(d(0))

Q is empty.

(1) 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:

f(f(x)) → f(c(f(x)))
f(f(x)) → f(d(f(x)))
g(c(x)) → x
g(d(x)) → x
g(c(0)) → g(d(1))
g(c(1)) → g(d(0))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1, 3, 5, 8, 11, 12, 15, 16, 18, 20, 22, 24, 25, 26

Node 1 is start node and node 3 is final node.

Those nodes are connected through the following edges:

  • 1 to 5 labelled f_1(0)
  • 1 to 11 labelled f_1(0)
  • 1 to 3 labelled f_1(0), c_1(0), d_1(0), g_1(0), 0(0), 1(0), f_1(1), c_1(1), d_1(1), g_1(1), 0(1), 1(1), 1(2), 0(2)
  • 1 to 15 labelled g_1(0)
  • 1 to 18 labelled f_1(1)
  • 1 to 22 labelled f_1(1)
  • 1 to 25 labelled g_1(1)
  • 3 to 3 labelled #_1(0)
  • 5 to 8 labelled c_1(0)
  • 8 to 3 labelled f_1(0)
  • 8 to 18 labelled f_1(1)
  • 8 to 22 labelled f_1(1)
  • 11 to 12 labelled d_1(0)
  • 12 to 3 labelled f_1(0)
  • 12 to 18 labelled f_1(1)
  • 12 to 22 labelled f_1(1)
  • 15 to 16 labelled d_1(0)
  • 16 to 3 labelled 1(0), 0(0)
  • 18 to 20 labelled c_1(1)
  • 20 to 3 labelled f_1(1)
  • 20 to 18 labelled f_1(1)
  • 20 to 22 labelled f_1(1)
  • 22 to 24 labelled d_1(1)
  • 24 to 3 labelled f_1(1)
  • 24 to 18 labelled f_1(1)
  • 24 to 22 labelled f_1(1)
  • 25 to 26 labelled d_1(1)
  • 26 to 3 labelled 1(1), 0(1)

(2) YES