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

0 QTRS
↳1 QTRS Reverse (⇔)
↳2 QTRS
↳3 RFCMatchBoundsTRSProof (⇔)
↳4 TRUE

(0) Obligation:

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

f(g(X)) → g(f(f(X)))
f(h(X)) → h(g(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:

g(f(x)) → f(f(g(x)))
h(f(x)) → g(h(x))

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:

g(f(x)) → f(f(g(x)))
h(f(x)) → g(h(x))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

9628812, 9628813, 9628814, 9628815, 9628816, 9628817, 9628818, 9628819

Node 9628812 is start node and node 9628813 is final node.

Those nodes are connect through the following edges:

  • 9628812 to 9628814 labelled f_1(0)
  • 9628812 to 9628816 labelled g_1(0)
  • 9628813 to 9628813 labelled #_1(0)
  • 9628814 to 9628815 labelled f_1(0)
  • 9628815 to 9628813 labelled g_1(0)
  • 9628815 to 9628818 labelled f_1(1)
  • 9628816 to 9628813 labelled h_1(0)
  • 9628816 to 9628817 labelled g_1(1)
  • 9628817 to 9628813 labelled h_1(1)
  • 9628817 to 9628817 labelled g_1(1)
  • 9628818 to 9628819 labelled f_1(1)
  • 9628819 to 9628813 labelled g_1(1)
  • 9628819 to 9628818 labelled f_1(1)

(4) TRUE