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

0 QTRS
↳1 RFCMatchBoundsTRSProof (⇔)
↳2 TRUE

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

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

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

9694038, 9694039, 9694041, 9694042, 9694040, 9694044, 9694043, 9694046, 9694045, 9694047, 9694048, 9694049, 9694050, 9694051, 9694053, 9694052

Node 9694038 is start node and node 9694039 is final node.

Those nodes are connect through the following edges:

  • 9694038 to 9694039 labelled f_1(0), h_1(0), 1(0), c_1(0), g_1(0), 0(0), d_1(0), f_1(1), h_1(1), 1(1), c_1(1), g_1(1), 0(1), d_1(1), 1(2)
  • 9694038 to 9694040 labelled g_1(0)
  • 9694038 to 9694043 labelled f_1(0)
  • 9694038 to 9694045 labelled f_1(0)
  • 9694038 to 9694051 labelled g_1(1)
  • 9694038 to 9694047 labelled f_1(1)
  • 9694038 to 9694049 labelled f_1(1)
  • 9694038 to 9694042 labelled h_1(1)
  • 9694038 to 9694053 labelled h_1(2)
  • 9694039 to 9694039 labelled #_1(0)
  • 9694041 to 9694042 labelled h_1(0)
  • 9694041 to 9694039 labelled 1(0)
  • 9694042 to 9694039 labelled 0(0)
  • 9694040 to 9694041 labelled d_1(0)
  • 9694044 to 9694039 labelled f_1(0)
  • 9694044 to 9694047 labelled f_1(1)
  • 9694044 to 9694049 labelled f_1(1)
  • 9694043 to 9694044 labelled c_1(0)
  • 9694046 to 9694039 labelled f_1(0)
  • 9694046 to 9694047 labelled f_1(1)
  • 9694046 to 9694049 labelled f_1(1)
  • 9694045 to 9694046 labelled d_1(0)
  • 9694047 to 9694048 labelled c_1(1)
  • 9694048 to 9694039 labelled f_1(1)
  • 9694048 to 9694047 labelled f_1(1)
  • 9694048 to 9694049 labelled f_1(1)
  • 9694049 to 9694050 labelled d_1(1)
  • 9694050 to 9694039 labelled f_1(1)
  • 9694050 to 9694047 labelled f_1(1)
  • 9694050 to 9694049 labelled f_1(1)
  • 9694051 to 9694052 labelled d_1(1)
  • 9694053 to 9694039 labelled 0(1)
  • 9694052 to 9694053 labelled h_1(1)
  • 9694052 to 9694039 labelled 1(1)

(2) TRUE