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:

7453119, 7453120, 7453121, 7453122, 7453124, 7453125, 7453123, 7453126, 7453127, 7453130, 7453129, 7453128, 7453131, 7453132, 7453133, 7453134, 7453136, 7453135, 7453137, 7453138, 7453139, 7453140, 7453141, 7453142

Node 7453119 is start node and node 7453120 is final node.

Those nodes are connect through the following edges:

  • 7453119 to 7453121 labelled g1_1(0)
  • 7453119 to 7453123 labelled f1_1(0)
  • 7453119 to 7453126 labelled f1_1(0)
  • 7453119 to 7453128 labelled g1_1(0)
  • 7453120 to 7453120 labelled #_1(0)
  • 7453121 to 7453122 labelled g_1(0)
  • 7453121 to 7453131 labelled g_1(1)
  • 7453122 to 7453120 labelled f_1(0)
  • 7453122 to 7453134 labelled f_1(1)
  • 7453124 to 7453125 labelled g_1(0)
  • 7453124 to 7453131 labelled g_1(1)
  • 7453125 to 7453120 labelled f_1(0)
  • 7453125 to 7453134 labelled f_1(1)
  • 7453123 to 7453124 labelled g_1(0)
  • 7453126 to 7453127 labelled f_1(0)
  • 7453126 to 7453134 labelled f_1(1)
  • 7453127 to 7453120 labelled g_1(0)
  • 7453127 to 7453131 labelled g_1(1)
  • 7453130 to 7453120 labelled g_1(0)
  • 7453130 to 7453131 labelled g_1(1)
  • 7453129 to 7453130 labelled f_1(0)
  • 7453129 to 7453134 labelled f_1(1)
  • 7453128 to 7453129 labelled f_1(0)
  • 7453131 to 7453132 labelled f_1(1)
  • 7453132 to 7453133 labelled f_1(1)
  • 7453132 to 7453134 labelled f_1(1)
  • 7453132 to 7453137 labelled f_1(2)
  • 7453133 to 7453120 labelled g_1(1)
  • 7453133 to 7453131 labelled g_1(1)
  • 7453134 to 7453135 labelled g_1(1)
  • 7453136 to 7453120 labelled f_1(1)
  • 7453136 to 7453134 labelled f_1(1)
  • 7453136 to 7453132 labelled f_1(1)
  • 7453135 to 7453136 labelled g_1(1)
  • 7453135 to 7453131 labelled g_1(1)
  • 7453135 to 7453140 labelled g_1(2)
  • 7453137 to 7453138 labelled g_1(2)
  • 7453138 to 7453139 labelled g_1(2)
  • 7453139 to 7453132 labelled f_1(2)
  • 7453140 to 7453141 labelled f_1(2)
  • 7453141 to 7453142 labelled f_1(2)
  • 7453142 to 7453135 labelled g_1(2)

(6) TRUE