YES Termination w.r.t. Q proof of Rubio_04_nestrec.ari

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:

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:

1, 4, 8, 9, 10, 11, 12, 13

Node 1 is start node and node 4 is final node.

Those nodes are connected through the following edges:

  • 1 to 8 labelled f_1(0)
  • 1 to 10 labelled g_1(0)
  • 4 to 4 labelled #_1(0)
  • 8 to 9 labelled f_1(0)
  • 9 to 4 labelled g_1(0)
  • 9 to 11 labelled f_1(1)
  • 10 to 4 labelled h_1(0)
  • 10 to 13 labelled g_1(1)
  • 11 to 12 labelled f_1(1)
  • 12 to 4 labelled g_1(1)
  • 12 to 11 labelled f_1(1)
  • 13 to 4 labelled h_1(1)
  • 13 to 13 labelled g_1(1)

(4) YES