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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 UsableRulesReductionPairsProof (⇔)
↳4 QDP
↳5 RFCMatchBoundsDPProof (⇔)
↳6 TRUE

(0) Obligation:

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

a(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, x))))

Q is empty.

(1) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(2) Obligation:

Q DP problem:
The TRS P consists of the following rules:

A(f, a(g, a(f, x))) → A(f, a(g, a(g, a(f, x))))
A(f, a(g, a(f, x))) → A(g, a(g, a(f, x)))
A(g, a(f, a(g, x))) → A(g, a(f, a(f, a(g, x))))
A(g, a(f, a(g, x))) → A(f, a(f, a(g, x)))

The TRS R consists of the following rules:

a(f, a(g, a(f, x))) → a(f, a(g, a(g, a(f, x))))
a(g, a(f, a(g, x))) → a(g, a(f, a(f, a(g, x))))

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(3) UsableRulesReductionPairsProof (EQUIVALENT transformation)

First, we A-transformed [FROCOS05] the QDP-Problem. Then we obtain the following A-transformed DP problem.
The pairs P are:

f1(g(f(x))) → f1(g(g(f(x))))
f1(g(f(x))) → g1(g(f(x)))
g1(f(g(x))) → g1(f(f(g(x))))
g1(f(g(x))) → f1(f(g(x)))

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

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

Q is empty.

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

No rules are removed from R.

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(f(x1)) = x1   
POL(f1(x1)) = 2·x1   
POL(g(x1)) = x1   
POL(g1(x1)) = 2·x1   

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

f1(g(f(x))) → f1(g(g(f(x))))
f1(g(f(x))) → g1(g(f(x)))
g1(f(g(x))) → g1(f(f(g(x))))
g1(f(g(x))) → f1(f(g(x)))

The TRS R consists of the following rules:

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

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

(5) RFCMatchBoundsDPProof (EQUIVALENT transformation)

Finiteness of the DP problem can be shown by a matchbound of 2.
As the DP problem is minimal we only have to initialize the certificate graph by the rules of P:

f1(g(f(x))) → f1(g(g(f(x))))
f1(g(f(x))) → g1(g(f(x)))
g1(f(g(x))) → g1(f(f(g(x))))
g1(f(g(x))) → f1(f(g(x)))

To find matches we regarded all rules of R and P:

g(f(g(x))) → g(f(f(g(x))))
f(g(f(x))) → f(g(g(f(x))))
f1(g(f(x))) → f1(g(g(f(x))))
f1(g(f(x))) → g1(g(f(x)))
g1(f(g(x))) → g1(f(f(g(x))))
g1(f(g(x))) → f1(f(g(x)))

The certificate found is represented by the following graph.

The certificate consists of the following enumerated nodes:

9608330, 9608331, 9608332, 9608333, 9608336, 9608334, 9608335, 9608337, 9608338, 9608339, 9608340, 9608341, 9608344, 9608343, 9608342, 9608345, 9608346, 9608347, 9608349, 9608350, 9608348, 9608351, 9608352, 9608353

Node 9608330 is start node and node 9608331 is final node.

Those nodes are connect through the following edges:

  • 9608330 to 9608332 labelled g1_1(0)
  • 9608330 to 9608334 labelled f1_1(0)
  • 9608330 to 9608337 labelled f1_1(0)
  • 9608330 to 9608339 labelled g1_1(0)
  • 9608331 to 9608331 labelled #_1(0)
  • 9608332 to 9608333 labelled g_1(0)
  • 9608332 to 9608345 labelled g_1(1)
  • 9608333 to 9608331 labelled f_1(0)
  • 9608333 to 9608342 labelled f_1(1)
  • 9608336 to 9608331 labelled f_1(0)
  • 9608336 to 9608342 labelled f_1(1)
  • 9608334 to 9608335 labelled g_1(0)
  • 9608335 to 9608336 labelled g_1(0)
  • 9608335 to 9608345 labelled g_1(1)
  • 9608337 to 9608338 labelled f_1(0)
  • 9608337 to 9608342 labelled f_1(1)
  • 9608338 to 9608331 labelled g_1(0)
  • 9608338 to 9608345 labelled g_1(1)
  • 9608339 to 9608340 labelled f_1(0)
  • 9608340 to 9608341 labelled f_1(0)
  • 9608340 to 9608342 labelled f_1(1)
  • 9608341 to 9608331 labelled g_1(0)
  • 9608341 to 9608345 labelled g_1(1)
  • 9608344 to 9608331 labelled f_1(1)
  • 9608344 to 9608342 labelled f_1(1)
  • 9608344 to 9608346 labelled f_1(1)
  • 9608343 to 9608344 labelled g_1(1)
  • 9608343 to 9608345 labelled g_1(1)
  • 9608343 to 9608351 labelled g_1(2)
  • 9608342 to 9608343 labelled g_1(1)
  • 9608345 to 9608346 labelled f_1(1)
  • 9608346 to 9608347 labelled f_1(1)
  • 9608346 to 9608342 labelled f_1(1)
  • 9608346 to 9608348 labelled f_1(2)
  • 9608347 to 9608331 labelled g_1(1)
  • 9608347 to 9608345 labelled g_1(1)
  • 9608347 to 9608343 labelled g_1(1)
  • 9608349 to 9608350 labelled g_1(2)
  • 9608350 to 9608346 labelled f_1(2)
  • 9608348 to 9608349 labelled g_1(2)
  • 9608351 to 9608352 labelled f_1(2)
  • 9608352 to 9608353 labelled f_1(2)
  • 9608353 to 9608343 labelled g_1(2)

(6) TRUE