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

0 QTRS
↳1 DependencyPairsProof (⇔)
↳2 QDP
↳3 QDPOrderProof (⇔)
↳4 QDP
↳5 RootLabelingFC2Proof (⇔)
↳6 QDP
↳7 DependencyGraphProof (⇔)
↳8 QDP
↳9 UsableRulesReductionPairsProof (⇔)
↳10 QDP
↳11 UsableRulesProof (⇔)
↳12 QDP
↳13 MNOCProof (⇔)
↳14 QDP
↳15 Narrowing (⇔)
↳16 QDP
↳17 Narrowing (⇔)
↳18 QDP
↳19 DependencyGraphProof (⇔)
↳20 TRUE

(0) Obligation:

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

f(y, f(x, y)) → f(f(a, y), f(a, 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:

F(y, f(x, y)) → F(f(a, y), f(a, x))
F(y, f(x, y)) → F(a, y)
F(y, f(x, y)) → F(a, x)

The TRS R consists of the following rules:

f(y, f(x, y)) → f(f(a, y), f(a, x))

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

(3) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].


The following pairs can be oriented strictly and are deleted.


F(y, f(x, y)) → F(a, y)
F(y, f(x, y)) → F(a, x)
The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:

POL(F(x1, x2)) = 0A + 1A·x1 + 1A·x2

POL(f(x1, x2)) = 5A + 1A·x1 + 1A·x2

POL(a) = 4A

The following usable rules [FROCOS05] were oriented:

f(y, f(x, y)) → f(f(a, y), f(a, x))

(4) Obligation:

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

F(y, f(x, y)) → F(f(a, y), f(a, x))

The TRS R consists of the following rules:

f(y, f(x, y)) → f(f(a, y), f(a, x))

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

(5) RootLabelingFC2Proof (EQUIVALENT transformation)

We used root labeling (second transformation) [ROOTLAB] with the following heuristic:
LabelAll: All function symbols get labeled
As Q is empty the root labeling was sound AND complete.

(6) Obligation:

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

F_{f,f}(y, f_{f,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
F_{f,f}(y, f_{a,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,a}(a, x))
F_{a,f}(y, f_{f,a}(x, y)) → F_{f,f}(f_{a,a}(a, y), f_{a,f}(a, x))
F_{a,f}(y, f_{a,a}(x, y)) → F_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))

The TRS R consists of the following rules:

f_{f,f}(y, f_{f,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
f_{f,f}(y, f_{a,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,a}(a, x))
f_{a,f}(y, f_{f,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,f}(a, x))
f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))

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

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(8) Obligation:

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

F_{f,f}(y, f_{f,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))

The TRS R consists of the following rules:

f_{f,f}(y, f_{f,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
f_{f,f}(y, f_{a,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,a}(a, x))
f_{a,f}(y, f_{f,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,f}(a, x))
f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))

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

(9) UsableRulesReductionPairsProof (EQUIVALENT transformation)

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.

The following rules are removed from R:

f_{a,f}(y, f_{f,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,f}(a, x))
Used ordering: POLO with Polynomial interpretation [POLO]:

POL(F_{f,f}(x1, x2)) = 2·x1 + x2   
POL(a) = 0   
POL(f_{a,a}(x1, x2)) = 2·x1 + x2   
POL(f_{a,f}(x1, x2)) = x1 + x2   
POL(f_{f,a}(x1, x2)) = 2·x1 + x2   
POL(f_{f,f}(x1, x2)) = 2·x1 + x2   

(10) Obligation:

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

F_{f,f}(y, f_{f,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))

The TRS R consists of the following rules:

f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))
f_{f,f}(y, f_{f,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))
f_{f,f}(y, f_{a,f}(x, y)) → f_{f,f}(f_{a,f}(a, y), f_{a,a}(a, x))

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

(11) UsableRulesProof (EQUIVALENT transformation)

We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.

(12) Obligation:

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

F_{f,f}(y, f_{f,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))

The TRS R consists of the following rules:

f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))

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

(13) MNOCProof (EQUIVALENT transformation)

We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.

(14) Obligation:

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

F_{f,f}(y, f_{f,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x))

The TRS R consists of the following rules:

f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))

The set Q consists of the following terms:

f_{a,f}(x0, f_{a,a}(x1, x0))

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

(15) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule F_{f,f}(y, f_{f,f}(x, y)) → F_{f,f}(f_{a,f}(a, y), f_{a,f}(a, x)) at position [1] we obtained the following new rules [LPAR04]:

F_{f,f}(y0, f_{f,f}(f_{a,a}(x1, a), y0)) → F_{f,f}(f_{a,f}(a, y0), f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, x1)))

(16) Obligation:

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

F_{f,f}(y0, f_{f,f}(f_{a,a}(x1, a), y0)) → F_{f,f}(f_{a,f}(a, y0), f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, x1)))

The TRS R consists of the following rules:

f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))

The set Q consists of the following terms:

f_{a,f}(x0, f_{a,a}(x1, x0))

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

(17) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule F_{f,f}(y0, f_{f,f}(f_{a,a}(x1, a), y0)) → F_{f,f}(f_{a,f}(a, y0), f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, x1))) at position [] we obtained the following new rules [LPAR04]:

F_{f,f}(f_{a,a}(x1, a), f_{f,f}(f_{a,a}(y1, a), f_{a,a}(x1, a))) → F_{f,f}(f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, x1)), f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, y1)))

(18) Obligation:

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

F_{f,f}(f_{a,a}(x1, a), f_{f,f}(f_{a,a}(y1, a), f_{a,a}(x1, a))) → F_{f,f}(f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, x1)), f_{f,f}(f_{a,a}(a, a), f_{a,a}(a, y1)))

The TRS R consists of the following rules:

f_{a,f}(y, f_{a,a}(x, y)) → f_{f,f}(f_{a,a}(a, y), f_{a,a}(a, x))

The set Q consists of the following terms:

f_{a,f}(x0, f_{a,a}(x1, x0))

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

(19) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(20) TRUE