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:

9694397, 9694398, 9694400, 9694399, 9694402, 9694401

Node 9694397 is start node and node 9694398 is final node.

Those nodes are connect through the following edges:

  • 9694397 to 9694399 labelled p_1(0), q_1(0)
  • 9694397 to 9694401 labelled p_1(0), q_1(0)
  • 9694398 to 9694398 labelled #_1(0)
  • 9694400 to 9694398 labelled f_1(0)
  • 9694399 to 9694400 labelled g_1(0)
  • 9694402 to 9694398 labelled g_1(0)
  • 9694401 to 9694402 labelled f_1(0)

(2) TRUE