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:

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

Q is empty.

(1) RFCMatchBoundsTRSProof (EQUIVALENT transformation)

Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 0. This implies Q-termination of R.
The following rules were used to construct the certificate:

p(f(f(x))) → q(f(g(x)))
p(g(g(x))) → q(g(f(x)))
q(f(f(x))) → p(f(g(x)))
q(g(g(x))) → p(g(f(x)))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

1355578, 1355579, 1355581, 1355580, 1355583, 1355582

Node 1355578 is start node and node 1355579 is final node.

Those nodes are connect through the following edges:

  • 1355578 to 1355580 labelled p_1(0), q_1(0)
  • 1355578 to 1355582 labelled p_1(0), q_1(0)
  • 1355579 to 1355579 labelled #_1(0)
  • 1355581 to 1355579 labelled f_1(0)
  • 1355580 to 1355581 labelled g_1(0)
  • 1355583 to 1355579 labelled g_1(0)
  • 1355582 to 1355583 labelled f_1(0)

(2) TRUE